Solution:

Substitute $x = 6$: $f(6) = 3(6) - 4$ $f(6) = 18 - 4$ $f(6) = 14$

Answer: $14$

Solution:

Substitute $x = -3$: $g(-3) = (-3)^2 + 2(-3)$ $g(-3) = 9 - 6$ $g(-3) = 3$

Answer: $3$

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

Solution:

Set up the equation: $h(a) = 17$ $2a + 5 = 17$

Solve: $2a = 12$ $a = 6$

Answer: $a = 6$

Check: $h(6) = 2(6) + 5 = 17$ ✓

Step 1: Replace every $x$ with $5$ in the function: $f(5) = 3(5) - 7$

Step 2: Evaluate: $f(5) = 15 - 7 = 8$

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 1: Set the function equal to zero: $x^2 - 4x + 3 = 0$

Step 2: Factor: $(x - 1)(x - 3) = 0$

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 1: Find $a$ using $g(a) = 13$: $2a + 1 = 13$ $2a = 12$ $a = 6$

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 1: Evaluate the inner function first — find $g(2)$: $g(2) = (2)^2 - 1 = 4 - 1 = 3$

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 $f$.

Step 1: We know $f(3) = 7$ (since the graph passes through $(3, 7)$).

Step 2: We need to find a value of $x$ where $g(x)$ can be evaluated. For $g(x) = f(2x - 1) + 3$, we need $2x - 1 = 3$ (so that $f$ is evaluated at 3, which we know).

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

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 $f$.