12 multiple-choice questions, progressively harder.
If f(x)=x2−4xf(x) = x^2 - 4xf(x)=x2−4x, what is f(4)f(4)f(4)?
Solution
Correct answer: B
Square the input, then subtract four times the input.
f(4)=42−4⋅4=16−16=0f(4) = 4^2 - 4 \cdot 4 = 16 - 16 = 0f(4)=42−4⋅4=16−16=0
For f(x)=x+4x2−1f(x) = \dfrac{x + 4}{x^2 - 1}f(x)=x2−1x+4, which inputs are excluded from the domain?
Correct answer: D
Exclude every input that makes the denominator zero.
x2−1=0 ⟹ x=1 or x=−1x^2 - 1 = 0 \;\Longrightarrow\; x = 1 \text{ or } x = -1x2−1=0⟹x=1 or x=−1
Both values are excluded; the numerator does not restrict the domain.
Which of these graphs would FAIL the vertical line test?
A graph fails when some vertical line meets it more than once. A circle has vertical lines that cross it twice.
one x ⟶ two y values on a circle\text{one } x \;\longrightarrow\; \text{two } y \text{ values on a circle}one x⟶two y values on a circle
The lines and the upward parabola each meet every vertical line only once, so they pass.
If f(x)=5f(x) = 5f(x)=5 for every input xxx (a constant function), what is f(100)f(100)f(100)?
Correct answer: A
A constant function returns the same output no matter the input.
f(100)=5f(100) = 5f(100)=5
If f(x)=5f(x) = 5f(x)=5 for every input, what is the range?
Correct answer: C
The range is the set of outputs actually produced, and this rule only ever produces 555.
range={5}\text{range} = \{5\}range={5}
Using the same function (f(1)=3f(1) = 3f(1)=3, f(2)=6f(2) = 6f(2)=6, f(3)=11f(3) = 11f(3)=11, f(4)=18f(4) = 18f(4)=18), what is f(4)−f(2)f(4) - f(2)f(4)−f(2)?
Read each output, then subtract.
f(4)−f(2)=18−6=12f(4) - f(2) = 18 - 6 = 12f(4)−f(2)=18−6=12
If f(x)=x3f(x) = x^3f(x)=x3, what is f(2)f(2)f(2)?
Cubing means multiply the input by itself three times.
f(2)=23=2⋅2⋅2=8f(2) = 2^3 = 2 \cdot 2 \cdot 2 = 8f(2)=23=2⋅2⋅2=8
The distractor 666 comes from tripling instead of cubing.
Let f(x)=x2−2f(x) = x^2 - 2f(x)=x2−2 and g(x)=x+5g(x) = x + 5g(x)=x+5. Which is larger, f(3)f(3)f(3) or g(3)g(3)g(3)?
Evaluate each function at 333 and compare.
f(3)=32−2=7,g(3)=3+5=8f(3) = 3^2 - 2 = 7, \qquad g(3) = 3 + 5 = 8f(3)=32−2=7,g(3)=3+5=8
Since 8>78 > 78>7, the value g(3)g(3)g(3) is larger.
If f(x)=x−2x+2f(x) = \dfrac{x - 2}{x + 2}f(x)=x+2x−2, which input is excluded from the domain?
Exclude the input that makes the denominator zero.
x+2=0 ⟹ x=−2x + 2 = 0 \;\Longrightarrow\; x = -2x+2=0⟹x=−2
So −2-2−2 is excluded; 222 is fine because it only makes the numerator zero.
A rule says: add 333 to the input, then double the result. Which function notation matches?
Do the steps in order: first x+3x + 3x+3, then double the whole thing.
f(x)=2(x+3)=2x+6f(x) = 2(x + 3) = 2x + 6f(x)=2(x+3)=2x+6
The rule 2x+32x + 32x+3 doubles before adding, which reverses the steps.
Using f(x)=2(x+3)f(x) = 2(x + 3)f(x)=2(x+3), what is f(−3)f(-3)f(−3)?
Add inside the parentheses first, then double.
f(−3)=2(−3+3)=2⋅0=0f(-3) = 2(-3 + 3) = 2 \cdot 0 = 0f(−3)=2(−3+3)=2⋅0=0
Which statement is true for EVERY function?
The definition of a function is about inputs: each input is tied to exactly one output.
each input ⟶ exactly one output\text{each input} \;\longrightarrow\; \text{exactly one output}each input⟶exactly one output
The other statements describe special kinds of functions, not all of them; for instance squaring sends 333 and −3-3−3 to the same output.
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