Objective 8 Linear Equations

Obj 8a Slope

The slope of the line through (x1, y1) and (x2, y2)

Gives the andof the slant of the line.

Find the slope of the line through:

(−4, 6) and (2,−7) (−4, 6) and (−4, 2) (3, 2) and (0, 2)

x

y

x

y

x

y

—————————————————————————————————————–

Obj 8e Writing the Equation of a line

How do we determine the algebraic rule that defines a particular line (the xy equation of a line)?

Each line is uniquely determined by and the .

The equation of the line through (x1, y1) with slope m is given byy − y1 = m(x− x1) Ax+By = C y = mx+ b

Obj 8eWrite an equation of the line that goes through (−1, 3) and has slopem = 2. Express answerin standard form. eGrade Instructions: Please remember that standard form is Ax+By = C, whereA, B, C are integers and A > 0. Enter the values A, B, C as an ordered 3-tuple (A,B,C).

1

Obj 8e Write an equation of the line that goes through (−6,−3) and has slope m = −5

7. Express

answer in standard form.

Obj 8e Write an equation of the line that goes through (−3, 9) and (12,−1). Express answer inslope-intercept form. eGrade Instructions: (Enter only the right hand side of the expression. Useproper or improper fractions or exact decimals; mixed fractions and decimal approximations arenot allowed.)

Obj 8e Write an equation of the line that goes through (0, b) and has slope m. Express answer inslope-intercept form.

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Obj 8c Identify the slope of a line.

Write in slope-intercept form. . (Unless special cases of vertical or horizontalline; see below.)

Find the slope of the line 2x− y = −5

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Obj 8b Graph of Linear Equation

***************Plan of Attack Just find the .It isn’t necessary to put in slope-intercept form.

2

Which of the following most closely resembles the graph of. 2x− y = −5

x

y

x

y

x

y

x

y

Obj 8b Which of the following most closely resembles the graph of. 11x+ 3y = −36

y

x

y

x

y

x

y

x

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Obj 8d Summary of facts for Vertical and Horizontal lines.

Horizontal line has equation , has slope , except for x-axis has .

Vertical line has equation , has slope , except for y-axis has .

Obj 8c Find the slope of the line:x = −4 x =

√3 y = 5 y = π

Obj 8d Write an equation of the line that goes through (−2, 4) that has undefined slope.

Obj 8d Write an equation of the line that goes through (−2, 4) that has no x-intercepts.

Obj 8d Write an equation of the line that goes through (−5, 1) and (−5, 4).

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Obj 8f Using slope of parallel or perpendicular lines to write an equation of a line.***************Plan of AttackFor nonvertical lines:Parallel lines have the .

Perpendicular lines have slopes whose .

For example, if a line has slopem = −3

5, then a line that’s perpendicular has slope .

For vertical lines, we must use geometry. Two vertical lines are parallel - they never intersect.A vertical line and a horizontal are perpendicular - they intersect in a 90◦ angle.***************

Obj 8f Write an equation of the line that goes through (−2, 5) that’s perpendicular to the line8x− 3y = −7. (The multiple choices are all in standard form, A,B,C, integers and A> 0.)

Obj 8f Write an equation of the line that goes through (4,−3) and is parallel to y =1

5x− 1.

Obj 8f Write an equation of the line that goes through (−1, 5) that’s parallel to the line x = 3.

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Obj 8e Linear Applications (Writing the equation of a line.)***************Plan of AttackWhen written in slope-intercept form, we say .

The “express” sentence tells you how to set up the ordered pairs. You cannot .

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As one descends into the ocean, pressure increases linearly. The pressure is 15 pounds per squareinch, on the surface, and 30 pounds per square inch, 33 feet below the surface. Express the pressurep (in pounds per square inch) in terms of the depth d (in feet below the surface). (Enter only theright hand side of the expression. Use correct variable. Use proper or improper fractions (no mixedfractions) or exact decimals. Do not give decimal approximations.)

Blue whales weigh approximately 2 tons when newborn. Young whales are nursed for 9 months,and by the time of weaning, they often weigh 25 tons. Let w denote the weight of the whale in tonsand let a be the age of the whale in months. If the weight is linearly related to the age, express win terms of a. (Same on-line cautions shown in first example.)

The volume v (in cubic centimeters) of a gas varies linearly with temperature t (in degrees Celsius).At a temperature of 12◦C a gas occupies a volume of 280cm3. When warmed to 66◦C, the gasoccupies a volume of 361cm3. Express v in terms of t. (Same eGrade cautions shown in firstexample.)

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The volume v (in cubic centimeters) of a gas varies linearly with temperature t (in degrees Celsius).At a temperature of 12◦C a gas occupies a volume of 280cm3. When warmed to 66◦C, the gasoccupies a volume of 361cm3. Express t in terms of v. (Same eGrade cautions shown in firstexample.)

A peanut farmer finds that at an average cost of $3 per bushel, 300 bushels can be produced; 1200bushels can be produced at an average of $2 per bushel. Assuming that the cost per bushel, c andthe number of bushels, n, are linearly related. Express c in terms of n.

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Obj 9 Center-Radius Form of the Equation of a Circle

x

y

Identify the center and radius.(x− 3)2 + (y − 2)2 = 25 (x− 2)2 + (y + 1)2 = 5 x2 + (y + 3)2 = 4

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Objective 10 Functions

A function is an algebraic rule (or a correspondence between sets) that associates each element, x,

from one set with a element, y, of another set.

The set of x’s is the .

The set of y’s is the .

Obj 10a Identify a Function (from graph or set of points)

Consider x2 + y2 = 4.The graph of a function must pass the

x

y

Which represent the graph of a function? (Read carefully; sometimes we ask “Which do notrepresent the graph of a function?”)

y

x

y

x

y

x

Which are functions?

{(1, 2), (3, 2), (5,−3)} {(1, 2), (1,−2), (−1, 3)}

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Obj 10b Function Notation; Evaluate a function

Instead of writing y = 3x2 − 5x+ 8, we now write f(x) = 3x2 − 5x+ 8

Instead of saying “let x = 2, find y”, we now write f(2)

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Find each of the following for the functions given below.

f(x) = 3x2 − 5x+ 8 g(x) = 3− x2 h(x) =| 3− 5x | m(x) = x n(x) = 2

f(x− 2) = g(−1) = g(2a) =

g(x+ h) = h(1) = h(−3) =

m(−3) = m(5) = n(−3) = n(5) =

m(2a) = m(x+ h) = n(2a) = n(x+ h) =

Obj 10c Domain and Range

Give Domain and Range (from the graph of a function.)

x

y

x

y

x

y

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More Obj 10c - Give Domain (only) from the function rule.

We’re being asked for the set of real numbers that can be substituted in for .

What prevents a value from being plugged in for x?When might the domain not be “all real numbers”?

There are only TWO situations that limit (restrict) the values that can be used for x:

Give the Domainf(x) = 3x7 + 5x4 +

√5 g(x) =| 3− 5x | +17 h(x) = 3

√1− x

f(x) =√3x+ 7

Some on-line problems ask: For f(x) =√3x+ 7 select the values that are in the domain of this

function.

Choices: −5 −1 0 1

Give the Domain f(x) = 4√2− 3x+ 7x

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Give the Domain f(x) =x+ 2

x− 3

Give the Domain f(x) =2x− 3

x2 − 4

Give the Domain f(x) =2x− 3

x2 + 4

Give the Domain f(x) =x+ 1

x2 − x− 6

Give the Domain f(x) =x+ 1

x2 + x− 1

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Some on-line problems just ask: Is the Domain “all reals” or “not all reals”?

When the denominator is quadratic, you can use the from the quadratic formulato answer.

Domain “all reals”? f(x) =x+ 1

x2 + x− 1

Domain “all reals”? f(x) =x+ 1

x2 + x+ 1

Domain “all reals”? f(x) =x− 3

| x+ 1 | f(x) =x

5− x− 3x2

Domain “all reals”? f(x) =5

x2 − x+ 6f(x) =

x

3+ 2x− 4

Domain “all reals”? f(x) =x+ 13√x− 2

f(x) =x+ 1√x− 2

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Obj 10d Find information from and about the graph of a function.

There are several different types of problems in this Objective.

Obj 10d example If f(x) =2x− 1

4x2, if the point (3, 1) on the graph of y = f(x)?

YES or NO

Obj 10d example For f(x) =x− 31

37− x, if f(x) = 2, what is x?

a) 35

b) −29

35c) none of these

Obj 10d examplesFor the function shown, For the function shown,for what x is f(x) > 0? for what x is f(x) < 0?

x

y

x

y

Obj 10d examplesFor the function shown, For the function shown,for what x is f(x) > 0? for what x is f(x) < 0?

x

y

x

y

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Obj 10d examplesFor the function shown, For the function shown,if x = 0, what is f(x)? for what x is f(x) = 0?

x

y

x

y

Obj 10d exampleAnswer the questions below for the function shown.

x

y

f(−1) is positive, negative, undefined

f(2) is positive, negative, undefined

f(2.5) is positive, negative, undefined

f(3) =

f(4) =

How many times does the line y = 5 intersect the graph?

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Obj 11 Properties of Functions

Obj 11a Increasing Decreasing Constant

Always look to (as x values are getting larger)

The function is if the y values are getting larger.

The function is if the y values are getting smaller.

The function is if the y values don’t change.

x

y(7,6)

(-3,-5)

-3-6-9 3 6 9 12

3

6

-3

-6

x

y

(6,-5)(-4,-5)

-3-6-9 3 6 9 12

3

6

-3

-6

Increasing on Increasing on

Decreasing on Decreasing on

Constant on Constant on

Obj 11b Local Max Local Min

A local is a point where the function changes from to .

A local is a point where the function changes from to .

The max/min IS the .

The max/min IS AT the .

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x

y

P(-2,4)

Q(2,-4)

R(7,8)

For point P: The local is . The local is at x = .

For point Q: The local is . The local is at x = .

For point R: The local is . The local is at x = .

Obj 11c Average Rate of Change

The average rate of change of f between c and x is the of the secant linebetween (c, f(c)) and (x, f(x)).

Average Rate of Change =

x

y

The average rate of change the slope of the graph at x = c.

In calculus, the derivative gives the slope of the graph at x = c, which is the slope of the tangentline at x = c.

***************Plan of Attack Set up the two ordered pairs (c, f(c)) and (x, f(x)) - fill in the values - then, justfind the slope. Always REDUCES.***************

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Obj 11c Example Find the average rate of change of f between −3 and x for f(x) = −3x2+2x−5.

Obj 11c Example Find the average rate of change of f between 4 and x for f(x) =−2

x− 1.

Obj 11c Example Find the average rate of change of f between −1 and x for f(x) = −x2 + 2x.

Obj 11c Example Find the average rate of change of f between 4 and x for f(x) =1

2− x.

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Obj 11c Example Find the average rate of change of f between 5 and x for f(x) = 2√x. (The

square root function examples will quiz only, not test.)

Obj 11c Example Find the average rate of change of f between 3 and x for f(x) =√x+ 1. (The

square root function examples will quiz only, not test.)

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Obj 11d Slope of the Secant Line

This is the basis of the definition of the derivative in calculus.(The Obj 11d problems will quiz only, not test.)

Find the slope of the secant line between (x, f(x)) and (x+ h, f(x+ h)) for f(x) = x2 + 2x− 3.

Find the slope of the secant line between (x, f(x)) and (x+ h, f(x+ h)) for f(x) = −x2 − 3.

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Obj 11e Symmetry Properties for Functions

x

y y

x

y

x

EVEN ODDGraph is symmetric Graph is symmetricwith respect to with respect to

f(−x) = f(−x) =

***************Plan of Attack To test for Symmetry, find f(−x)

***************

Obj 11e Even/Odd/Neither

f(x) = x4 − 3x2 + 5 f(x) = x7 − 5x5 + 3x

f(x) = x5 − 5x+ 2 f(x) =2x3

2x2 − 3

f(x) =2

x5− 1

x+

1

2f(x) =

x+ 1

x2 − 4

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Obj 12 Collection of 7 basic functions

***************Plan of Attack You DO NOT need to memorize the properties - Learn the GRAPH and theFUNCTION rule. You should be able to answer any question, if you know the GRAPH.***************1) Special case of a linear function f(x) = x.

x

y

2) Squaring function f(x) = x2.

x

y

3) Cubing function f(x) = x3.

x

y

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4) Square Root function f(x) =√x.

x

y

5) Cube Root function f(x) = 3√x.

x

y

6) Absolute value function f(x) =| x |.

x

y

7) Reciprocal function f(x) =1

x.

x

y

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Obj 12 example

Enter the equation of the function whose graph is shown. (Enter both sides of the equation, notjust the right hand side. You must use y on the left side of the equation; f(x) is not allowed in thismode.)

x

y

Your Answer

Obj 12 exampleSelect all of the following that are symmetric with respect to the y-axis. (There may be more thanone.)

f(x) = x f(x) = x2 f(x) = x3

f(x) =√x f(x) = 3

√x f(x) =| x | f(x) =

1

x

Obj 12 exampleSelect all of the following that are true for the function f(x) =| x |. (There may be more than one.)

Domain is (−∞,∞) Domain is [0,∞) Domain is (−∞, 0) ∪ (0,∞)

Range is (−∞,∞) Range is [0,∞) Range is (−∞, 0) ∪ (0,∞)

Graph is symmetric with respect to the y-axis. Graph is symmetric with respect to the origin.

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Obj 13 Functions defined piecewise

A function defined piecewise has a different function rule for different parts of the domain.

Obj 13a Evaluate a functions defined piecewise

***************Plan of Attack Compare the x-value with the restrictions on the right - that tells you whichfunction rule to use. DO NOT plug x into each!***************

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f(x) =

x+ 2, if x ≤ 3x2, if 3 < x ≤ 50, if x > 5

Find each of the following

f(−1) = f(3) =

f(3.01) = f(3.5) = f(5) =

f(5.001) = f(12) =

f(x) =

−1, if x ≤ −10, if − 1 < x < 11, if x ≥ 1

Find each of the following

f(−1) = f(0.5) =

Obj 13b Graph a functions defined piecewise

A warm-up questionGraph f(x) = x− 2 if x ≥ 1 Graph f(x) = x− 2 if x > 1

x

y

x

y

22

Graph f(x) ={

x− 1, if x ≥ 35− 2x, if x < 3

x

y

Graph f(x) ={

3x− 1, if x < −1−4, if x ≥ −1

x

y

Graph f(x) ={

4− 3x, if x ≥ 02x+ 4, if x < 0

x

y

23

Graph f(x) ={

3x− 5, if x < 2−2, if x ≥ 2

x

y

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Obj 14 Graphing with Reflections, Compressions/Stretching, Translations

Obj 14a Graphing with Reflections

How do these compare?y = f(x) y = −f(x) y = f(−x)

We will consider a specific example to justify the general case.y =

√x y = −√

x y =√−x

x

y

x

y

x

y

24

For y = f(x) as defined below, graph y = −f(x) and y = f(−x).

y

x1

x

y

x

y

another Obj 14a example If (−3, 0) is a point on the graph of y = f(x), then which of thefollowing must be on the graph of y = f(−x)?

(0, 3) (0,−3) (3, 0) (−3, 0)

another Obj 14a example If (−3, 0) is a point on the graph of y = f(x), then which of thefollowing must be on the graph of y = −f(x)?

(0, 3) (0,−3) (3, 0) (−3, 0)

another Obj 14a example If a function f has Domain [−2, 0], then what must be the Domainof y = −f(x)?

[0, 2] [−2, 0] [−2, 2]

another Obj 14a example If a function f has Domain [−2, 0], then what must be the Domainof y = f(−x)?

[0, 2] [−2, 0] [−2, 2]

another Obj 14a example If a function f has Range [−2, 0], then what must be the Range ofy = −f(x)?

[0, 2] [−2, 0] [−2, 2]

another Obj 14a example If a function f has Range [−2, 0], then what must be the Range ofy = f(−x)?

[0, 2] [−2, 0] [−2, 2]

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Obj 14c Graphing with Vertical and Horizontal Translations

How do these compare?y = f(x) y = f(x) + c, c > 0 y = f(x)− c, c > 0

We will consider a specific example to justify the general case.

y =√x y =

√x+ 3 y =

√x− 3

x

y

x

y

x

y

How do these compare?y = f(x) y = f(x+ c), c > 0 y = f(x− c), c > 0

We will consider a specific example to justify the general case.

y =√x y =

√x+ 3 y =

√x− 3

x

y

x

y

x

y

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Obj 14c example Select the function whose graph is the graph of y = x5 but is shifted left 3 units.

y = x5 − 3 y = (x+ 3)5 y = (x− 3)5 y = x5 + 3

Obj 14c example For y = f(x) as defined below, graph y = f(x)− 2 and y = f(x− 2).

y

x1

x

y

x

y

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Obj 14b Graphing with Vertical or Horizontal Compression or Stretching

How do these compare?y = f(x) y = cf(x), c > 1 y = cf(x), 0 < c < 1

We will consider a specific example to justify the general case.

y =√4− x2 y = 2

√4− x2 y = 1

2

√4− x2

x

y

x

y

x

y

27

How do these compare?y = f(x) y = f(cx), c > 1 y = f(cx), 0 < c < 1

We will consider a specific example to justify the general case.

y =√4− x2 y =

√

4− (2x)2 y =√

4− (12x)2

x

y

x

y

x

y

For y = f(x) as defined below, graph y = 3f(x) and y = f(13x).

y

x1

x

y

x

y

28

another Obj 14b example If (2, 4) is a point on the graph of y = f(x), then which of the followingmust be on the graph of y = f(2x)?

(4, 4) (4, 8) (1, 2) (1, 4) (2, 4) (2, 8)

another Obj 14b example If (4, 0) is a point on the graph of y = f(x), then which of the followingmust be on the graph of y = 2f(x)?

(4, 0) (8, 0) (2, 0) (0, 4) (0, 8) (0, 2)

another Obj 14b example If a function f has Domain [0, 8], then what must be the Domain ofy = 4f(x)?

[0, 32] [0, 2] [0, 8]

another Obj 14b example If a function f has Domain [0, 8], then what must be the Domain ofy = f(4x)?

[0, 32] [0, 2] [0, 8]

another Obj 14b example If a function f has Range [0, 8], then what must be the Range ofy = 4f(x)?

[0, 32] [0, 2] [0, 8]

another Obj 14b example If a function f has Range [0, 8], then what must be the Range ofy = f(4x)?

[0, 32] [0, 2] [0, 8]

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Obj 14 Summary

Affects Range Affects Domain

Obj 14a y = −f(x) y = f(−x)

Obj 14b y = 3f(x) y = f(3x)

Obj 14b y = 1

4f(x) y = f(1

4x)

Obj 14c y = f(x) + 2 y = f(x+ 2)

Obj 14c y = f(x)− 5 y = f(x− 5)

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Obj 15 Operations on Functions - forming new functions by adding, subtracting, multiplying, ordividing existing functions

Definition of notation used to denote these functions; defined for all x in the domain of f and g.

(f + g)(x) = (f − g)(x) =

(fg)(x) =

(

f

g

)

(x) =

Obj 15a examples

f(x) =x2 + 1

x+ 2g(x) = 3

√5− 2x h(x) = −3 m(x) = x

Find (g −m)(16). (Integer or exact decimal only; mathematical operators are not allowed.)

Find (m+ h)(0). (Integer or exact decimal only; mathematical operators are not allowed.)

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Find

(

f

h

)

(4). (If non-integer answer, give fraction or exact decimal. Don’t give decimal approxi-

mation.)

Find (hf)(3). (If non-integer answer, give fraction or exact decimal. Don’t give decimal approxi-mation.)

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Obj 15b examples Find the domain of a sum, difference, product, or quotient function. Recallthat these functions are defined for all x in the domain of f and g.

For f(x) =√2x+ 8 and g(x) =

√−x− 2, find the domain of f − g.

For f(x) =√1− x and g(x) =

√1− 4x, find the domain of fg.

For f(x) =√x− 4 and g(x) =

√−x− 2, find the domain of f + g.

For f(x) =√3x− 9 and g(x) =

x

2x− 5, find the domain of f + g.

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For f(x) =√−3x− 9 and g(x) =

x

2x− 5, find the domain of f − g.

For f(x) = 3√3x− 9 and g(x) =

x

2x− 5, find the domain of f − g.

For f(x) =1

3√3x− 9

and g(x) =x

2x− 5, find the domain of f + g.

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Obj 16 Function Composition

Given two functions f and g, then the composite function, denoted as , is defined asfollows

(f ◦ g)(x) = for all x in the domain of g such that g(x) is in the domain of f .

Obj 16a examples

f(x) =1

1− xg(x) = 3x2 − 2x

Find (f ◦ g)(−2) Find (g ◦ f)(−2)

32

Find (f ◦ g)(x) Find (g ◦ g)(x)

Find (f ◦ f)(x)

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Obj 16b examples

For f(x) =x

x− 2and g(x) =

3

x− 3,

Give the domain of f ◦ g. Give the domain of g ◦ f

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For f(x) =1

xand g(x) =

1

x− 2,

Give the domain of f ◦ g. Give the domain of g ◦ f

For f(x) =2x

x− 5and g(x) =

6x

x− 2,

Give the domain of f ◦ g. Give the domain of g ◦ f

For f(x) = x2 and g(x) =√x = 4,

Give the domain of f ◦ g.

Copyright c©2010-present, Annette Blackwelder, all rights are reserved. Reproduction or distribu-tion of these notes is strictly forbidden.

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