Algebra 2 Practice Test: Piecewise Functions
- Piecewise Functions
- 10 questions
- 15 minutes
These 10 questions practice evaluating piecewise functions, writing absolute value as a piecewise function, step functions, continuity at a boundary and real-world pricing rules. For every input, first decide which piece applies by checking the condition.
Tap an option to mark your answer. You can change your mind, or tap the same option again to leave the question blank. The correct answers and explanations appear when you finish the test.Tap an option to answer; tap it again to leave the question blank.
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Question 1
f(x) = x2 − 1 if x < 2
f(x) = 3x − 4 if x ≥ 2
What is f(2)?
Correct answer: A
Since 2 ≥ 2, use the second piece: f(2) = 3 · 2 − 4 = 2. Using the first piece gives 22 − 1 = 3, but that piece is only for x < 2.
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Question 2
f(x) = x2 − 1 if x < 2
f(x) = 3x − 4 if x ≥ 2
What is f(−3)?
Correct answer: B
Since −3 < 2, use the first piece: (−3)2 − 1 = 9 − 1 = 8. The second piece would give −13, and writing −32 as −9 gives −10.
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Question 3
Which piecewise definition is equal to g(x) = |x − 3|?
Correct answer: A
When x < 3, x − 3 is negative, so the absolute value is −(x − 3) = 3 − x. The pieces change at x = 3, where x − 3 = 0, not at x = 0.
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Question 4
h(x) = 2x + k if x < 1
h(x) = x2 + 4 if x ≥ 1
For which value of k is the graph of h connected (continuous) at x = 1?
Correct answer: B
Both pieces must give the same value at x = 1: 2 · 1 + k = 12 + 4 = 5, so k = 3. The value 5 is the height where the pieces meet, not k.
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Question 5
A courier charges $6 for a package up to 2 pounds and $6 plus $2 for each pound over 2 pounds for heavier packages, with part-pounds charged proportionally. What does a 5.5-pound package cost?
Correct answer: C
The package is 3.5 pounds over the limit, so the cost is 6 + 2 · 3.5 = $13. Charging $2 for all 5.5 pounds on top of $6 gives $17.
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Question 6
The greatest integer function ⌊x⌋ gives the greatest integer less than or equal to x. What is ⌊−2.3⌋?
Correct answer: D
The integers less than or equal to −2.3 are −3, −4, −5, … and the greatest of these is −3. Dropping the decimal gives −2, which is greater than −2.3.
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Question 7
f(x) = x + 4 if x ≤ 0
f(x) = 2x − 1 if x > 0
What is the solution set of f(x) = 5?
Correct answer: B
First piece: x + 4 = 5 gives x = 1, but this piece needs x ≤ 0, so 1 is rejected. Second piece: 2x − 1 = 5 gives x = 3, which satisfies x > 0.
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Question 8
f(x) = −x if x < 0
f(x) = 2 if x ≥ 0
What is the range of f?
Correct answer: D
For x < 0, −x takes every positive value, and the second piece adds only the value 2, which is already positive. The value 0 is never reached, because x = 0 uses the second piece.
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Question 9
A parking garage charges $4 for the first hour or part of an hour and $3 for each additional hour or part of an hour. How much does it cost to park for 4 hours 10 minutes?
Correct answer: C
The stay starts a fifth hour, so you pay for the first hour plus 4 additional hours: 4 + 4 · 3 = $16. Counting only 3 additional hours gives $13.
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Question 10
f(x) = x + 2 if x < 0
f(x) = −x + 2 if x ≥ 0
What is the maximum value of f?
Correct answer: A
The left piece rises toward 2 and the right piece starts at f(0) = 2 and falls, so the graph is an upside-down V with its peak at (0, 2). This function equals 2 − |x|.
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