evaluating functions with ti 83 84
TRANSCRIPT
TC3 → Stan Brown → TI-83/84/89 → Evaluating Functions
Summary:
Contents:
See also:
revised 5 Aug 2007
Evaluating Functions with TI-83/84
Copyright © 2001–2012 by Stan Brown, Oak Road Systems
Very often, you need to evaluate a function for several values of x. Examples: �inding a
limit or an approximation numerically, plotting on paper by plotting points. This page
gives you two techniques, creating a table of values and tracing a graph. We’ll use this
function to show the methods:
f(x) = (x³ − 1) / (x − 1).
Both Methods: Enter the Function
Method 1: Table of Values
Method 2: Trace on the Graph
Graphing Functions on TI‐83/84
Graphing Piecewise Functions on TI‐83/84
Both Methods: Enter the Function
You have to tell the TI‐83/84 what your function is, before you can use Method 1 or Method 2.
If any other plots are active
(highlighted), you need to turn them off.
If nothing is highlighted on your Y=
screen, you can skip this step.
It so happens that I had Plot1 and Y2
active, as you can see below left. In the
right‐hand panel I’ve deactivated both
of them.
Press [Y=].
Use the blue arrow keys to move the cursor onto every
highlighted = sign or Plot number, and press [ENTER]. Your
screen will look different, but that’s okay as long as you have
nothing highlighted.
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Enter the function. Cursor to any empty Y= slot.
Press [(] [x,T,θ,n] [^] [3] [−] 1 [)] [÷] [(] [x,T,θ,n] [−] 1 [)]
[ENTER].
You may be using a different function number, but make sure your entry
looks like my Y3. If you made a mistake, go back and edit it.
Method 1: Table of Values
This method works strictly by numbers.
Press [2nd WINDOW makes TBLSET].
The �irst two rows don’t matter. Press [▼] [▼] to get to the “Indpnt” row. This controls the
independent variable x.
Press [►] [ENTER] to select “Ask.”
Press [▼] [ENTER] to select “Depend: Auto.”
Your top two rows may be different, but your bottom two rows will look
like the screen at right. These settings tell the TI‐83/84 to ask you for
values of the independent variable x, then automatically calculate the
values of the dependent variable f(x).
Now you can evaluate the function at
selected x values.
Press [2nd GRAPH makes TABLE].
You may see some values on the table screen. They don’t do
any harm, but if you want you can get rid of them by hitting
[DEL] several times.
Enter the x values, one at a time. For
instance, to home in on the limit as x
approaches 1, we might enter .5, .75, .9,
.99, and so on.
Enter each x number and press [ENTER]. The TI‐83/84
immediately displays the function value.
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Method 2: Trace on the Graph
This method is a little more work, but you get a picture of the function. Start by entering the function on
the Y= screen as shown above.
Set formatting so that the desired
information will appear on your graph.
Press [2nd ZOOM makes FORMAT].
The other settings are not critical, but you need CoordOn and
ExprOn. Press the arrow keys and [ENTER] to set the modes.
Either set up the Window screen, or use
“Zoom Standard” for a �irst look at the
graph. It happens that we want the limit
as x goes to 1. Since that �its within the
standard window, we’ll use Zoom
Standard this time.
Press [ZOOM] [6] to select ZoomStd. The graph should appear
(below left).
Press [TRACE], then an x value, then [ENTER]. You should see
the function, the x value, and the y value displayed (below
right).
Enter any other values, such as .99, .995,
.999. The dot will move along the graph,
and the new y values will be displayed.
You don’t have to press [TRACE] again. Simply enter each new
x value, followed by [ENTER].
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For this function it’s not an issue, but for other functions if your x value is outside the window, you need
to press [WINDOW] and adjust Xmin or Xmax. You can only trace x values that are between Xmin and
Xmax.
This page is used in instruction at Tompkins Cortland Community College in Dryden, New York; it’s not an of�icialstatement of the College. Please visit www.tc3.edu/instruct/sbrown/ to report errors or ask to copy it.
For updates and new info, go to http://www.tc3.edu/instruct/sbrown/ti83/
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TC3 → Stan Brown → TI-83/84/89 → Graphing Functions
Summary:
See also:
Contents:
revised 5 Aug 2007
Graphing Functions on TI-83/84
Copyright © 2001–2012 by Stan Brown, Oak Road Systems
It’s pretty easy to produce some kind of graph on the TI‐83/84 for a given function. This
page helps you with the tricks that might not be obvious. You’ll be able to "ind
asymptotes, intercepts, intersections, roots, and so on.
Evaluating Functions with TI‐83/84
Graphing Piecewise Functions on TI‐83/84
Graphing Your Function
Common Problems
Tuning Your Graph
Zooming
Adjusting the Window
Adjusting the Grid
Exploring Your Graph
Domain and Asymptotes
Function Values
Intercepts
Multiple Functions
Intersection
The techniques in this note will work with any function, but for purposes of illustration, we’ll use
Graphing Your Function
Step 1: Clear unwanted plots.
You need to look for any previously set
plots that might interfere with your
new one.
Press [Y=] (the top left button). Look at the top of the screen.
If any of Plot1 Plot2 Plot3
is highlighted, cursor to it and
press [ENTER] to deactivate it.
(No information is lost; you
can always go back and
reactivate any plot.) To verify
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that you have deactivated the
plot, cursor away from it and check that it’s not highlighted.
(Sometimes you might want to graph
more than one function on the same
axis. In this case, make sure to
deactivate all the functions you don’t
want to graph.)
Now check the lines starting with Y1=, Y2=, and so on. If any =
sign is highlighted, either delete the whole equation or
deactivate it but leave it in memory. To delete an equation,
cursor to it and press the [CLEAR] button. To deactivate it
without deleting it, cursor to its = sign and press [ENTER].
My screen looked like this
after I deactivated all old plots
and functions.
Step 2: Enter the function.
If your function is not already in y=
form, use algebra to transform it before
proceeding.
Two cautions:
For x, use the [x,T,θ,n] key,
not the [×] (times) key.
The TI‐83/84 follows the
standard order of operations.
If there are operations on top
or bottom of a fraction, you
must use parentheses — for
x+2 divided by x−3, you can’t
just enter “x+2/x−3”.
Cursor to one of the Y= lines, press [CLEAR] if necessary, and
enter the function.
Check your function and correct any
mistakes.
For example, if you see a star * in place
of an X, you accidentally used the times
key instead of [x,T,θ,n].
Use the [◄] key and overtype any mistakes.
To delete any extra characters, press [DEL].
If you need to insert characters, locate yellow INS above the
[DEL] key. Press [2nd DEL makes INS] and type the additional
characters. As soon as you use a cursor key, the TI‐83/84
goes back to overtype mode.
Step 3: Display the graph.
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“Zoom Standard” is usually a good
starting point. It selects standard
parameters of ‐10 to +10 for x and y.
Press [ZOOM] [6].
Common Problems
If you don’t see your function graph anywhere, your window is probably restricted to a region of the xy
plane the graph just doesn’t happen to go through. Depending on the function, one of these techniques
will work:
ZoomFit is a good "irst try. Press [ZOOM] [0]. (Thanks to Marilyn Webb for this suggestion.)
You can try to zoom out (like going higher to see more of the xy plane) by pressing [ZOOM] [3]
[ENTER].
Finally, you can directly adjust the window to select a speci"ic region.
For other problems, please see TI‐83/84 Troubleshooting.
Tuning Your Graph
You can make lots of adjustments to improve your view of the function graph.
Zooming
The window is your "ield of view into the xy plane, and there are two main ways to adjust it. This
section talks about zooming, which is easy and covers most situations. The next section talks about
manually adjusting the window parameters for complete "lexibility.
Here’s a summary of the zooming techniques you’re likely to use:
You’ve already met standard zoom, which is [ZOOM] [6]. It’s a good starting point for most
graphs.
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You’ve also met zoom "it, which is [ZOOM] [0]. It slides the view "ield up or down to bring the
function graph into view, and it may also stretch or shrink the graph vertically.
To zoom out, getting a larger "ield of view with less detail, press [ZOOM] [3] [ENTER]. You’ll see
the graph again, with a blinking zoom cursor. You can press [ENTER] again to zoom out even
further.
To zoom in, focusing in on a part of the graph with more detail, press [ZOOM] [2] but don’t press
[ENTER] yet. The graph redisplays with a blinking zoom cursor in the middle of the screen. Use
the arrow keys to move the zoom cursor to the part of the graph you want to focus on, and
then press [ENTER]. After the graph redisplays, you still have a blinking zoom cursor and you
can move it again and press [ENTER] for even more detail.
Your viewing window is rectangular, not square. When your x and y axes have the same
numerical settings the graph is actually stretched by 50% horizontally. If you want a plot
where the x and y axes are to the same scale, press [ZOOM] [5] for square zoom.
There are still more variations on zooming. Some long winter evening, you can read about them in the
manual.
Adjusting the Window
You may want to adjust the window parameters to see more of the graph, to focus in on just one part, or
to get more or fewer tick marks. If so, press [WINDOW].
Xmin and Xmax are the left and right edges of the window.
Xscl controls the spacing of tick marks on the x axis. For instance, Xscl=2 puts tick marks
every 2 units on the x axis. A bigger Xscl spaces the tick marks farther apart, and a smaller
Xscl places them closer together.
Ymin and Ymax are the bottom and top edges of the window.
Yscl spaces the tick marks on the y axis.
If you want to blow up a part of the graph for a more detailed view, increase Xmin or Ymin or both, or
reduce Xmax or Ymax. Then press [GRAPH].
If you want to see more of the xy plane, compressed to a smaller scale, reduce Xmin and/or Ymin, or
increase Xmax or Ymax. Then press [GRAPH].
Many of the graph windows shown in your textbook will have small numbers printed at the four edges.
If you want to make your graphing window look like the one in the textbook, press use the numbers at
left and right edges for Xmin and Xmax, the number at the bottom edge for Ymin, and the number at the
top edge for Ymax.
Adjusting the Grid
The grid is the dots over the whole window that line up to the tick marks on the axes, kind of like graph
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paper. The grid helps you see the coordinates of points on the graph.
If you see a lot of horizontal lines running across the graph, your Xscl is way too small, and the tick
marks are running together in lines. Similarly, Yscl is the number of y units between tick marks. A
bunch of vertical lines means your Yscl is too small. Press [WINDOW] and "ix either of these problems.
To turn the grid on or off: Locate yellow FORMAT above
the [ZOOM] key. Press [2nd ZOOM
makes FORMAT].
Cursor to the desired GridOn
or GridOff setting, and press
[ENTER] to lock it in.
Then press [GRAPH] to return to your graph.
Exploring Your Graph
Domain and Asymptotes
First off, just look at the shape of the graph. A vertical asymptote should
stick out like a sore thumb, such as x = 3 with this function. (Con"irm
vertical asymptotes by checking the function de"inition. Putting x = 3 in
the function de"inition makes the denominator equal zero, which tells
you that you have an asymptote.)
The domain certainly excludes any x values where there are
vertical asymptotes. But additional values may also be excluded, even if
they’re not so obvious. For instance, the graph of f(x) = (x³+1)/(x+1) looks like a simple parabola, but
the domain does not include x = −1.
Horizontal asymptotes are usually obvious. But sometimes an apparent asymptote really isn’t
one, just looks like it because your "ield of view is too small or too large. Always do some algebra work
to con"irm the asymptotes. This function seems to have y = 1 as a horizontal asymptote as x gets very
small or very large, and in fact from the function de"inition you can see that that’s true.
Function Values
While displaying your graph, press [TRACE] and then the x value you’re interested in. The TI‐83/84 will
move the cursor to that point on the graph, and will display the corresponding y value at the bottom.
The x value must be within the current viewing window. If you get the message ERR:INVALID, press
[1] for Quit. Then adjust your viewing window and try again.
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Intercepts
You can trace along the graph to "ind any intercept. The intercepts of a graph are where it crosses or
touches an axis:
x intercept where graph crosses or touches x axis because y = 0
y intercept where graph crosses or touches y axis because x = 0
Most often it’s the x intercepts you’re interested in, because the x intercepts of the graph y = f(x) are the
solutions to the equation f(x) = 0, also known as the zeroes of the function.
To "ind x intercepts: You could naïvely press [TRACE] and cursor left and right, zooming in to make a
closer approximation. But it’s much easier to make the TI‐83/84 "ind the intercept for you.
Locate an x intercept by eye. For
instance, this graph seems to have an x
intercept somewhere between x = −3
and x = −1.
Locate yellow CALC above the [TRACE] key. Press [2nd TRACE
makes CALC] [2]. (You select 2:zero because the x intercepts
are zeroes of the function.)
Enter the left and right bounds. [(-)] 3 [ENTER] [(-)] 1 [ENTER]
There’s no need to make a
guess; just press [ENTER]
again.
Two cautions with x intercepts:
Since the TI‐83/84 does approximations, you must always check the TI‐83/84 answer in the
function de"inition to make sure that y comes out exactly 0.
When you "ind x intercepts, make sure to "ind all of them. This particular function has only one
in its entire domain, but with other functions you may have to look for additional x intercepts
outside the viewing area.
Finding the y intercept is even easier: press [TRACE] 0 and read off the y
intercept.
This y intercept looks like it’s about −2/3, and by plugging x = 0 in
the function de"inition you see that the intercept is exactly −2/3.
Multiple Functions
You can plot multiple functions on the same screen. Simply press [Y=] and enter the second function
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next to Y2=. Press [GRAPH] to see the two graphs together.
To select which function to trace along, press [▲] or [▼]. The upper left corner shows which function
you’re tracing.
Intersection
When you graph multiple functions on the same set of axes, you can have the TI‐83/84 tell you where
the graphs intersect. This is equivalent to solving a system of equations graphically.
The naïve approach is to trace along one graph until it crosses the other, but again you can do
better. We’ll illustrate by "inding the intersections of y =(6/5)x−8 with the function we’ve already
graphed.
Graph both functions on the same set of
axes. Zoom out if necessary to "ind all
solutions.
Press [2nd TRACE makes CALC] [5].
You’ll be prompted First curve? If necessary, press [▲] or
[▼] to select one of the curves you’re interested in. Press
[ENTER].
You’ll be prompted Second curve? If necessary, press [▲] or
[▼] to select the other curve you’re interested in. Press
[ENTER].
Eyeball an approximate solution. For
instance, in this graph there seems to be
a solution around x = 2.
When prompted Guess?, enter
your guess. In this case, since
your guess is 2 you should
press 2 [ENTER].
Repeat for any other solutions.
As always, you should con"irm apparent solutions by substituting in both equations. The TI‐83/84 uses
a method of successive approximations, which may create an ugly decimal when in fact there’s an exact
solution as a fraction or radical.
This page is used in instruction at Tompkins Cortland Community College in Dryden, New York; it’s not an of"icialstatement of the College. Please visit www.tc3.edu/instruct/sbrown/ to report errors or ask to copy it.
For updates and new info, go to http://www.tc3.edu/instruct/sbrown/ti83/
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TC3 → Stan Brown → TI-83/84/89 → Piecewise Functions
Summary:
See also:
revised 4 Oct 2008
Graphing Piecewise Functions on TI-83/84
Copyright © 2003–2012 by Stan Brown, Oak Road Systems
You can graph piecewise functions on your TI‐83/84 by using the TEST menu. To show
the method, we’ll graph the function
which is read “f of x equals x²+11 for x<0, 11−4x for x between 0 and 2 inclusive, and
x²−3x+5 for x>2.” This particular function, as you’ll see, doesn't have any gaps in it, but
exactly the same technique works for piecewise functions that do have gaps.
Graphing Functions on TI‐83/84
Set-up: Dot Mode
The TI‐83/84 likes to connect dots with
continuous lines or curves where it can.
But a piecewise function could have
gaps legitimately, and therefore you
want to select dot mode.
The TI‐83 and TI‐84 MODE screens are
slightly different, but the settings are
the same.
[MODE] [▼ 4 times] [►] [ENTER]
(You may need to switch between dot mode and connected mode, depending on the functions you’re
graphing, because a function with a steeply sloping graph will be hard to see in dot mode.)
Enter the Function
The general form you’re going for is
(=irst piece)(=irst condition)+(second piece)(second condition)+...
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This works because in the TI programming language a true condition is equivalent to a 1 and a false
condition to a zero. Therefore each branch of the function is turned on (multiplied by 1) in the proper
region and turned off (multiplied by 0) everywhere else.
You can have as many (piece)(condition) pairs as it takes to de=ine the function, and you always
need the parentheses around each piece and around each condition. If you have a compound condition
like 0 ≤ x ≤ 2, you can use [2nd MATH makes TEST] [►] [1] to create an and condition, or code the two
conditions inparentheses and multiply them.
For our sample function, you want to get this onto the Y= screen:
Y1=(x²+11)(x<0)+(11−4x)(0≤x and x≤2)+(x²−3x+5)(x>2)
or
Y1=(x²+11)(x<0)+(11−4x)(0≤x)(x≤2)+(x²−3x+5)(x>2)
You already know how to do all of that except the inequality signs in the tests, and as you’ll see, that’s
pretty easy.
Clear any previous plots. (Review this
on the general graphing page if you
need to.)
[Y=] and deactivate anything that’s highlighted.
Enter the =irst branch of the function
de=inition, (x²+11).
On the Y= screen, cursor to one of the Y= lines. Press [CLEAR]
if necessary, and enter the =irst piece in parentheses:
[(] [x,T,θ,n] [x²] [+] 11 [)]
Enter the test, (x<0). Press [(] [x,T,θ,n] [2nd MATH
makes TEST] [5] 0 [)]
Enter the second branch of the function
de=inition, (11−4x).
[+] [(] 11 [−] 4 [x,T,θ,n] [)]
Enter the second test, (0 ≤ x ≤ 2). You
can code this either as the product of
two tests, (0≤x)(x≤2), or with an and
condition, (0≤x and x≤2). The =irst way
saves a couple of keystrokes, so that’s
what I’ll do.
[(] 0 [2nd MATH makes TEST] [6]
[x,T,θ,n] [)] [(] [x,T,θ,n]
[2nd MATH makes TEST] [6] 2 [)]
Enter a plus sign and the last branch of
the function, (x²−3x+5).
[+] [(] [x,T,θ,n] [x²] [−] 3 [x,T,θ,n] [+] 5 [)]
Enter the last test, (x>2). [(] [x,T,θ,n] [2nd MATH makes
TEST] [3] 2 [)]
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Display the Graph
It’s often helpful to start with [ZOOM] [6], standard zoom, and then adjust the window. This particular
function, I think, is a little easier to visualize with the window parameters shown.
You can zoom, trace, and =ind values and intercepts just as you would do for any other function.
See the general graphing page for common problems.
One particular problem with piecewise functions is that the TI‐83/84 may try to connect the
pieces. Make sure you are in dot mode, not connected mode: look on the Y= screen for three dots to the
left of your equation.
What’s New
4 Oct 2008: change the function from two to three parts, to illustrate all three types of
conditions; explain how it works; explain compound conditions; add TI‐84 mode screen
26 Oct 2007: clarify TI‐83/84 keystrokes involving the [2nd] key
(various formatting improvements, suppressed)
31 Mar 2003: new document
This page is used in instruction at Tompkins Cortland Community College in Dryden, New York; it’s not an of=icialstatement of the College. Please visit www.tc3.edu/instruct/sbrown/ to report errors or ask to copy it.
For updates and new info, go to http://www.tc3.edu/instruct/sbrown/ti83/
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TC3 → Stan Brown → TI-83/84/89 → Inverse Functions
Summary:
revised 15 Feb 2009 (What’s New?)
Inverse Functions on TI-83/84
Copyright © 2001–2012 by Stan Brown, Oak Road Systems
Graph an Inverse Function
After you graph a function on your TI‐83/84, you can make a picture of its inverse byusing the DrawInv command on the DRAW menu.
For this illustration, let’s use f(x) = √(x−2), shown at right. Though youcan easily $ind the inverse of this particular function algebraically, thetechniques on this page will work for any function.
I’ve compensated for the rectangular viewing window by settingwindow margins to 0 to 10 in the x direction and 0 to 6.5 in the ydirection. (If you don’t know how to graph a function, please review thatprocedure.)
The graph of an inverse function is a mirror image of the original through the line y=x, so I’ve havealso plotted that line.
To draw the inverse of that function:
Paste the DrawInv command to yourhome screen.
[2nd PRGM makes DRAW] Either cursor down to the 8 and press [ENTER], or simplypress [8].
Tell the TI‐83/84 to $ind the originalfunction in Y1.
Press [VARS] [►] [1] [1].(If your function was in a different numbered y variable, pickthat one instead of Y1.)
At this point your screen shows thiscommand: DrawInv Y1
Now execute the command. Press [ENTER].
The result is shown at right.
You know from your algebra work that the inverse off(x) = √(x−2)
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isf (x) = x²+2, x ≥ 0
and the graph con$irms that.
Each point on the graph of f(x) has a corresponding point on the graph of f (x). For example, f(2) = 0,so (2,0) is on the original graph, (0,2) is on the graph of f (x), and f (0) = 2.
Unfortunately, all you can do with the inverse is look at it. You can’t trace or do other things. But eventhat helps you check your work. For instance, you see that the inverse of the sample function appearsonly in the positive x region. The inverse you calculate algebraically, x²+2, has a domain in both thepositive and negative reals, but from drawing the inverse on the TI‐83/84 you can see that you need torestrict the inverse function’s domain to match the restricted range of the original function.
There’s another way you can check your work. Find the inverse function $irst, algebraically, and graph itas Y3 when you graph the original as Y1. If you do that, DrawInv Y1 will exactly overlay the graph ofyour algebraic inverse.
Caution: Because the screen resolution is low, two different functions sometimes look the same.This method isn’t an absolute guarantee that your work is correct, but it’s better than no check at all.
Find the Value of an Inverse Function
Now suppose you have to $ind f (1.5)? Of course you can look at it on the graph and estimate, but yourcalculator can do a better job of the estimation for you. There are two methods, one on the graph andone on the home screen.
Method 1: intersect on Graph
Remember that f (1.5) is some value, call it a, such that (1.5,a) is on thegraph of f (x), and therefore (a,1.5) is on the graph of f(x). In otherwords, f (1.5) is the x value on the original graph of f(x) where the yvalue is 1.5.
Using this idea, to $ind f (1.5) you can plot y = 1.5 and have yourcalculator $ind the point where it intersects the graph of f(x). You don’tneed the graph of f (x) for this at all.
The graphs are shown at right, and here’s the procedure.
Select the intersect command. [2nd TRACE makes CALC] [5]
The calculator asks “First curve?” Simply press [ENTER] to select the $irst curve.
‐1
‐1
‐1 ‐1
‐1
‐1
‐1
‐1
‐1
‐1
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The calculator then asks “Secondcurve?”
The cursor may have moved automatically to the othercurve. If not, press [▲] or [▼] until it does. Press [ENTER].
Finally, the calculator asks for yourguess.
Usually you can just press [ENTER]. But if the function is verycomplicated, you can use [◄] or [►] to move the cursor closeto the intersection point and then press [ENTER], or type in anumber and press [ENTER].
The result is shown at right: the answer is 4.25.Why is the answer x and not y? Because you’re trying to $ind f (1.5),
the value of the inverse function of 1.5. But as mentioned above, f (1.5)is the number a such that f(a) = 1.5. In other words, becausef(4.25) = 1.5, f (1.5) = 4.25.
Caution: Your calculator gives numerical solutions only. To determinewhether 4.25 is the exact answer or just a good approximation, you haveto check it in the original function.
Method 2: solve on Home Screen
You can accomplish the same thing on the home screen by using the solve function.
Select the solve function from thecatalog because it’s not in a menu.(There’s a Solver command in the Mathmenu, but setting it up is a little morework.)
Press [2nd 0 makes CATALOG] [ALPHA 4 makes T], scroll up tosolve(, and press [ENTER].
The $irst argument is an expression thatyou want to equate to zero. You actuallywant to equate Y1 to 1.5, which is thesame as equating Y1−1.5 to 0.
Press [VARS] [►] [1] [1] [−] [1] [.] [5]
The second argument is the variable, x. Press [,] [x,T,θ,n].
‐1
-1
‐1
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The last argument is your initial guess.Unless the function is prettycomplicated, it doesn’t matter what youenter here as long as it’s in the domainof the function. For example, 0 would bea bad choice for f(x) = √(x−2) becausef(0) is not a real number. Let’s use 6 asthe initial guess.
Enter the initial guess and a close parenthesis [)].
The screen is shown at right. The answer of 4.25 agrees with thegraphical method.
Caution: Again, remember that this is a numerical solution andmay not be exact.
What’s New
15 Feb 2009: Add section, suggested by Max Harwood, on $inding value of inverse function; edit thegraphing section extensively for clarity; drop “drawing” from document title
(intervening changes suppressed)
7 Jun 2003: $irst publication, as “Drawing Inverse Functions on the TI‐83”
This page is used in instruction at Tompkins Cortland Community College in Dryden, New York; it’s not an of$icialstatement of the College. Please visit www.tc3.edu/instruct/sbrown/ to report errors or ask to copy it.
For updates and new info, go to http://www.tc3.edu/instruct/sbrown/ti83/
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