Interpreting Function Notation
$f(x)$ is read "$f$ of $x$" — the output of the function for a given input. The SAT tests whether you can read it as input-to-output.
Reading the concept isn't enough. The score comes from practicing the real Bluebook-style format and finding the exact mistakes you keep making.
What the SAT tests here
$f(5) = 11$ means: when the input is $5$, the output is $11$. To evaluate, substitute the input wherever $x$ appears.
How to solve one, step by step
Example: $f(x) = 3x - 4$. Find $f(5)$.
- Substitute $5$ for $x$: $3(5) - 4$.
- That's $11$, so $f(5) = 11$.
The mistakes that cost points
- Misreading which number is the output. $f(3) = 7$ means input $3$, output $7$ — not $x = 7$.
- Splitting f(a + b). $f(a + b)$ is generally not $f(a) + f(b)$.
Practice questions
Try these the way you would on test day, then open the solution to check your method.
Easy
If $f(x) = 7x - 22$, what is the value of $f(2) - f(t)$?
- A$-7t - 8$
- B$-7t - 14$
- C$-7t + 24$
- D$-7t + 14$
Show solution
Answer: D, $-7t + 14$. First evaluate $f(2)$ by substituting $2$ for $x$: $f(2) = 7(2) - 22 = 14 - 22$. Then evaluate $f(t)$ by substituting $t$ for $x$: $f(t) = 7(t) - 22 = 7t - 22$. Subtract: $f(2) - f(t) = (-8) - (7t - 22) = -7t + 14$.
Medium
If $g(x) = -4x + 5$, what is the value of $g(w - 9)$?
- A$-4w - 31$
- B$-4w + 5$
- C$-4w + 36$
- D$-4w + 41$
Show solution
Answer: D, $-4w + 41$. Substitute $(w - 9)$ for $x$ in the function: $g(w - 9) = -4(w - 9) + 5$. Distribute: $-4w + 36 + 5$. Combine the constants: $-4w + 41$.
Hard
The linear function $g$ is defined by $g(x) = k - 6x$, where $k$ is a constant. If $g(c + 6) = \frac{c}{2}$, where $c$ is a constant, which of the following represents the value of $k$ in terms of $c$?
- A$3c + 36$
- B$\frac{7c}{2} + 36$
- C$6c + 36$
- D$\frac{13c}{2} + 36$
Show solution
Answer: D, $\frac{13c}{2} + 36$. Substitute $(c + 6)$ for $x$ in the function: $g(c + 6) = k - 6(c + 6)$. Set equal to $\frac{c}{2}$: $k - 6(c + 6) = \frac{c}{2}$. Distribute and isolate $k$, then combine terms in $c$ over a common denominator. The result is $k = \frac{13c}{2} + 36$.
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