Unit 3 of 5

Unit 3: Functions and Their Graphs

Study guide for CLEP CLEP College Algebra — Unit 3: Functions and Their Graphs. Practice questions, key concepts, and exam tips.

56

Practice Questions

63

Flashcards

10

Key Topics

Key Concepts to Study

domain and range

function notation

transformations

inverse functions

graphing

piecewise functions

scenario-to-graph reasoning (real-world piecewise modeling)

graph interpretation from verbal description

step functions

evaluating piecewise functions

Sample Practice Questions

Try these 5 questions from this unit. Sign up for full access to all 56.

Q1HARD

A company's profit from selling x units of a product is given by the function P(x) = $2x^{2}$ + 5x - 100. What is the behavior of the function over the interval where x > 5?

A.

The function is increasing over the interval.

B.

The function is decreasing over the interval.

C.

The function is constant over the interval.

D.

The function is undefined over the interval.

E.

The function is increasing then decreasing over the interval

Show Answer

Answer: A — Since the coefficient of $x^{2}$ is positive, the parabola opens upward. For x > 5, as x increases, P(x) also increases, indicating the function is increasing.

Q2MEDIUM

Find the equation of the axis of symmetry for the parabola f(x) = $x^{2}$ - 4x + 3.

A.

x = 1

B.

x = 2

C.

x = 0

D.

x = -2

E.

x = -1

Show Answer

Answer: B — x = 2 is correct because x = -b / 2a = 4 / 2 = 2.

Q3MEDIUM

Which equation represents a graph that is a vertical shift of y = $x^{2}$ ?

A.

y = $x^{2}$ - 2

B.

y = $(x + 2)^{2}$

C.

y = $(x - 2)^{2}$

D.

y = $2x^{2}$

E.

y = $x^{2}$ + 2

Show Answer

Answer: E — "y = $x^{2}$ + 2" is correct because it shifts the parabola up by 2 units, while C and D are wrong since they represent horizontal shifts.

Q4MEDIUM

If f(x) = 2x + 1, which of the following is the inverse function $f^{-1}$ (x)?

A.

$f^{-1}$ (x) = x/2 - 1

B.

$f^{-1}$ (x) = x/2 + 1

C.

$f^{-1}$ (x) = (x + 1)/2

D.

$f^{-1}$ (x) = (x - 1)/2

E.

$f^{-1}$ (x) = 2x - 1

Show Answer

Answer: D — To find the inverse, swap x and y, then solve for y: x = 2y + 1, x - 1 = 2y, (x - 1)/2 = y.

Q5HARD

Consider the functions f(x) = $x^{2}$ and g(x) = 2^x. What is the effect on the domain of the composite function (f ∘ g)(x) compared to the domain of g(x)?

A.

The domain of (f ∘ g)(x) includes only positive real numbers.

B.

The domain of (f ∘ g)(x) excludes zero

C.

The domain of (f ∘ g)(x) includes all real numbers except where g(x) is undefined.

D.

The domain of (f ∘ g)(x) includes only negative real numbers.

E.

The domain of (f ∘ g)(x) is the same as the domain of g(x).

Show Answer

Answer: E — Since f(x) is defined for all real numbers and g(x) is also defined for all real numbers, (f ∘ g)(x) = f(g(x)) = (2^x)^2 is defined for all real numbers.

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Study Tips for Unit 3: Functions and Their Graphs

Focus on understanding concepts, not memorizing facts — CLEP tests application
Practice with timed questions to build exam-day speed
Review explanations for wrong answers — they reveal common misconceptions
Use flashcards for key terms, practice questions for deeper understanding