12 multiple-choice questions, progressively harder.
If f(x)=3x−2f(x) = 3x - 2f(x)=3x−2, what is f(a)+f(1)f(a) + f(1)f(a)+f(1)?
Solution
Correct answer: B
Evaluate each piece and add. Here f(a)=3a−2f(a) = 3a - 2f(a)=3a−2 and f(1)=3(1)−2=1f(1) = 3(1) - 2 = 1f(1)=3(1)−2=1.
f(a)+f(1)=(3a−2)+1=3a−1f(a) + f(1) = (3a - 2) + 1 = 3a - 1f(a)+f(1)=(3a−2)+1=3a−1
A taxi charges a base fee of 333 dollars plus 222 dollars for each mile. Writing the cost as a function of the number of miles mmm, which rule is correct?
Correct answer: A
The cost is the per-mile charge times the miles, plus the fixed base fee.
f(m)=2m+3f(m) = 2m + 3f(m)=2m+3
The 222 multiplies the miles and the 333 is added once.
What is the domain of f(x)=1x−3f(x) = \dfrac{1}{x - 3}f(x)=x−31?
The only forbidden input makes the denominator zero.
x−3=0 ⟹ x=3x - 3 = 0 \;\Longrightarrow\; x = 3x−3=0⟹x=3
So the domain is all numbers except 333.
Using the taxi rule f(m)=2m+3f(m) = 2m + 3f(m)=2m+3, what is the cost for a 444 mile trip?
Correct answer: C
Substitute 444 for mmm.
f(4)=2⋅4+3=8+3=11f(4) = 2 \cdot 4 + 3 = 8 + 3 = 11f(4)=2⋅4+3=8+3=11
So the trip costs 111111 dollars.
A function has f(1)=3f(1) = 3f(1)=3, f(2)=3f(2) = 3f(2)=3, and f(3)=3f(3) = 3f(3)=3. What is its range?
The range is the set of outputs the function actually produces, listing each value once.
range={3}\text{range} = \{3\}range={3}
Even though three inputs are used, the only output produced is 333.
If f(x)=∣x−5∣f(x) = |x - 5|f(x)=∣x−5∣, what is f(2)f(2)f(2)?
Correct answer: D
Subtract inside the bars first, then take the absolute value.
f(2)=∣2−5∣=∣−3∣=3f(2) = |2 - 5| = |-3| = 3f(2)=∣2−5∣=∣−3∣=3
Absolute value cannot be negative, so the distractor −3-3−3 is wrong.
For f(x)=5xf(x) = \dfrac{5}{x}f(x)=x5, can the output f(x)f(x)f(x) ever equal 000?
A fraction equals zero only when its numerator is zero, but the numerator here is always 555.
5x=0 has no solution\frac{5}{x} = 0 \;\text{has no solution}x5=0has no solution
For very large xxx the output gets close to 000 but never reaches it.
A function is given by f(1)=3f(1) = 3f(1)=3, f(2)=6f(2) = 6f(2)=6, f(3)=11f(3) = 11f(3)=11, and f(4)=18f(4) = 18f(4)=18. What is f(3)f(3)f(3)?
Read the output paired with the input 333.
f(3)=11f(3) = 11f(3)=11
If f(x)=xx−1f(x) = \dfrac{x}{x - 1}f(x)=x−1x, what is f(0)f(0)f(0)?
Check that 000 is in the domain (the denominator is −1-1−1, not zero), then evaluate.
f(0)=00−1=0−1=0f(0) = \frac{0}{0 - 1} = \frac{0}{-1} = 0f(0)=0−10=−10=0
If f(x)=x+1f(x) = \sqrt{x} + 1f(x)=x+1, what is f(9)f(9)f(9)?
Take the square root first, then add 111.
f(9)=9+1=3+1=4f(9) = \sqrt{9} + 1 = 3 + 1 = 4f(9)=9+1=3+1=4
If f(x)=x2f(x) = x^2f(x)=x2 and the output is f(x)=49f(x) = 49f(x)=49, what are all the inputs xxx?
Solve x2=49x^2 = 49x2=49 for every input that works.
x2=49 ⟹ x=7 or x=−7x^2 = 49 \;\Longrightarrow\; x = 7 \text{ or } x = -7x2=49⟹x=7 or x=−7
Two inputs share the output 494949, which a function allows.
If f(x)=4x−1f(x) = 4x - 1f(x)=4x−1, which input xxx makes f(x)=0f(x) = 0f(x)=0?
Set the rule equal to 000 and solve for the input.
4x−1=0 ⟹ 4x=1 ⟹ x=144x - 1 = 0 \;\Longrightarrow\; 4x = 1 \;\Longrightarrow\; x = \tfrac{1}{4}4x−1=0⟹4x=1⟹x=41
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