Function Notation on the SAT
What I tell my students first
$f(x)$ is not $f$ times $x$. It is the output of $f$ when the input is $x$. So $f(4)$ means: put $4$ in place of every $x$ and work out the result.
Then ask yourself which side the question gives you. "Find $f(4)$" gives you the input and asks for the output. "$f(a) = 19$, find $a$" gives you the output and asks for the input, so you set up an equation and solve it.
Function notation shows up all over SAT Math, in plain algebra, in word problems and in graphs. The math inside is usually not hard. When students lose points it is almost always because they mixed up the input and the output, or substituted into only some of the $x$'s.
Four examples
Example 1: input given, find the output
If $g(x) = x^2 - 2x + 1$, what is $g(-3)$?
- Put $-3$ in parentheses everywhere $x$ appears: $g(-3) = (-3)^2 - 2(-3) + 1$.
- $(-3)^2 = 9$ and $-2(-3) = 6$, so $g(-3) = 9 + 6 + 1 = 16$.
Example 2: output given, find the input
If $f(x) = 2x + 7$ and $f(a) = 19$, what is $a$?
- $f(a) = 2a + 7$, so $2a + 7 = 19$.
- $2a = 12$, and $a = 6$.
- Check: $f(6) = 12 + 7 = 19$.
Example 3: finding a constant
For $f(x) = ax^2 + 5$, $f(2) = 17$. What is $f(-3)$?
- Use the known value: $a(2)^2 + 5 = 17$, so $4a = 12$ and $a = 3$.
- Now the rule is $f(x) = 3x^2 + 5$, and $f(-3) = 3(9) + 5 = 32$.
Example 4: an expression as the input
If $f(x + 2) = 3x - 1$ for all $x$, what is $f(5)$?
- The input here is $x + 2$, so ask: what $x$ makes $x + 2 = 5$? It is $x = 3$.
- Put $x = 3$ into the right side: $3(3) - 1 = 8$. So $f(5) = 8$.
- Putting $5$ straight into $3x - 1$ gives $14$, which is $f(7)$, not $f(5)$.
What you will see
- $f(4)$. Replace every $x$ with $4$, in parentheses, and evaluate.
- $f(a) = 19$. Set the rule equal to $19$ and solve for $a$.
- $f(2a)$ or $f(x + 1)$. Replace every $x$ with the whole expression, in parentheses, then simplify.
- A constant in the rule, plus one known value. Substitute the known input and output to find the constant first.
- $f(g(2))$. Work from the inside out: find $g(2)$, then put that number into $f$.
- "What does $p(25) = 340$ mean?" The input is $25$ of something and the output is $340$ of something else. Name both with their units.
Mistakes I see often
- Reading $f(x)$ as multiplication. $f(3)$ is the output for input $3$, not $3f$.
- Leaving out parentheses for negatives. With $f(x) = x^2$, $f(-3) = (-3)^2 = 9$. Without parentheses you get $-9$.
- Solving when you should evaluate, or the reverse. "Find $f(19)$" and "$f(a) = 19$" are different questions. Check whether $19$ is the input or the output.
- Substituting into only part of the rule. For $f(2a)$ with $f(x) = x^2 + 4$, the answer is $(2a)^2 + 4 = 4a^2 + 4$. Writing $2a^2 + 4$ squares only the $a$.
- Working $f(g(x))$ from the outside in. Always start with the inner function.
Practice questions
Before you start each one, decide whether the question gives you an input or an output.
Easy
If $f(x) = 3x - 5$, what is the value of $f(4)$?
- A$12$
- B$7$
- C$17$
- D$3$
Show solution
Answer: B. $f(4) = 3(4) - 5 = 12 - 5 = 7$. Choice C adds $5$ instead of subtracting it, and choice D solves $f(x) = 4$ instead of finding $f(4)$.
EasyStudent-produced response
If $g(x) = x^2 - 2x + 1$, what is the value of $g(-3)$?
Enter your answer.
Show solution
Answer: $16$. $g(-3) = (-3)^2 - 2(-3) + 1 = 9 + 6 + 1 = 16$.
Medium
The function $f$ is defined by $f(x) = 2x + 7$. If $f(a) = 19$, what is the value of $a$?
- A$45$
- B$13$
- C$6$
- D$\dfrac{19}{2}$
Show solution
Answer: C. $2a + 7 = 19$, so $2a = 12$ and $a = 6$. Choice A is $f(19)$, which treats $19$ as the input instead of the output.
Medium
If $f(x) = x^2 + 4$, which expression is equal to $f(2a)$?
- A$2a^2 + 4$
- B$4a^2 + 4$
- C$2a^2 + 8$
- D$4a^2 + 16$
Show solution
Answer: B. Replace $x$ with $2a$: $(2a)^2 + 4 = 4a^2 + 4$. Choice A squares only the $a$, and choice C is $2f(a)$, which doubles the output instead of the input.
MediumStudent-produced response
The function $h$ is defined by $h(x) = kx - 3$, where $k$ is a constant. If $h(5) = 17$, what is the value of $k$?
Enter your answer.
Show solution
Answer: $4$. $h(5) = 5k - 3 = 17$, so $5k = 20$ and $k = 4$.
Medium
The function $f$ is defined by $f(x) = \dfrac{x + 9}{x - 3}$. Which of the following values of $x$ is not in the domain of $f$?
- A$-9$
- B$-3$
- C$3$
- D$9$
Show solution
Answer: C. A fraction is undefined when its denominator is $0$. $x - 3 = 0$ when $x = 3$. At $x = -9$ the numerator is $0$, which is allowed: $f(-9) = 0$.
HardStudent-produced response
The functions $f$ and $g$ are defined by $f(x) = x^2 - 3x$ and $g(x) = 2x + 1$. What is the value of $f(g(2))$?
Enter your answer.
Show solution
Answer: $10$. Start inside: $g(2) = 2(2) + 1 = 5$. Then $f(5) = 25 - 15 = 10$.
Hard
For the function $f$, $f(x + 2) = 3x - 1$ for all values of $x$. What is the value of $f(5)$?
- A$14$
- B$11$
- C$16$
- D$8$
Show solution
Answer: D. The input is $x + 2$. For the input to be $5$, $x$ must be $3$. Then $f(5) = 3(3) - 1 = 8$. Choice A puts $5$ straight into $3x - 1$, which gives $f(7)$.
HardStudent-produced response
The function $f$ is defined by $f(x) = ax^2 + 5$, where $a$ is a constant. If $f(2) = 17$, what is the value of $f(-3)$?
Enter your answer.
Show solution
Answer: $32$. $4a + 5 = 17$, so $a = 3$. Then $f(-3) = 3(-3)^2 + 5 = 27 + 5 = 32$.
Hard
A print shop uses the function $p(n) = 12n + 40$ to find the cost, in dollars, of printing $n$ T-shirts. What is the best interpretation of $p(25) = 340$?
- APrinting $340$ T-shirts costs \$25.
- BPrinting $25$ T-shirts costs \$340.
- CEach T-shirt costs \$25 to print.
- DPrinting $25$ more T-shirts adds \$340 to the cost.
Show solution
Answer: B. The input $n = 25$ is the number of T-shirts and the output $340$ is the cost in dollars. So printing $25$ T-shirts costs \$340. Choice A swaps the input and the output.
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