Types of SAT Function Questions

1. Evaluation

"If $f(x) = x^2 - 3$, what is $f(4)$?"

Answer: $f(4) = 16 - 3 = 13$

2. Composition

$(f \circ g)(x) = f(g(x))$

If $f(x) = 2x$ and $g(x) = x + 1$: $f(g(3)) = f(4) = 8$

3. Finding Input

"If $f(x) = 3x - 1$ and $f(a) = 14$, what is $a$?"

Solve: $3a - 1 = 14 \Rightarrow a = 5$

Domain and Range

• Domain: All possible input values (x-values)
• Range: All possible output values (y-values)

SAT Function Tricks

Watch for:

• $f(x + 2)$ vs $f(x) + 2$ (very different!)
• Questions asking for $2f(3)$ when they give you $f(3) = 5$
  • Answer: $2(5) = 10$, not $f(6)$!

Answer: $3$

SAT Tip: Be careful with negatives! $(-3)^2 = 9$

Answer: $a = 6$

Check: $h(6) = 2(6) + 5 = 17$ ✓

Answer: $f(5) = 8$

Key concept: $f(5)$ means "plug in 5 for $x$." Function notation is just a way to name the output.

Step 3: Apply zero product property: $x - 1 = 0 \implies x = 1$ $x - 3 = 0 \implies x = 3$

Check: $f(1) = 1 - 4 + 3 = 0$ ✓ and $f(3) = 9 - 12 + 3 = 0$ ✓

Answer: $x = 1$ and $x = 3$

SAT Context: These are the $x$-intercepts (zeros/roots) of the function's graph.

Step 2: Find $g(2a) = g(12)$: $g(12) = 2(12) + 1 = 25$

Answer: $g(2a) = 25$

Alternative approach: Notice $g(2a) = 2(2a) + 1 = 4a + 1$. Since $2a = 12$, we get $g(2a) = 2(12) + 1 = 25$.

Step 2: Now evaluate $f(g(2)) = f(3)$: $f(3) = 2(3) + 3 = 9$

Answer: $f(g(2)) = 9$

Key concept: Composition of functions — work from the inside out. First evaluate $g(2)$, then plug that result into $f$.

$2x - 1 = 3$ $2x = 4$ $x = 2$

Step 3: Calculate $g(2)$: $g(2) = f(2(2) - 1) + 3 = f(3) + 3 = 7 + 3 = 10$

Answer: The point $(2, 10)$ lies on the graph of $y = g(x)$.

SAT Tip: For transformation questions, figure out what input to $g$ produces a known input to $f$.