According to College Board specifications, these topics fall under "Passport to Advanced Math," which accounts for a significant portion of the scoring weight. Many of these problems also overlap with Hard SAT Algebra Practice Questions, as solving for $x$ is often the final step in a function-based word problem.

2. **Solved Examples**

Example 1: Composite Functions
If $f(x) = 3x^2 - 5$ and $g(x) = 2x + 1$, what is the value of $f(g(2))$?

1. First, evaluate the inner function: $g(2) = 2(2) + 1 = 5$.
2. Substitute this result into the outer function: $f(5) = 3(5)^2 - 5$.
3. Calculate the square: $5^2 = 25$.
4. Multiply and subtract: $3(25) - 5 = 75 - 5 = 70$.
5. The final answer is 70.

Example 2: Function Transformations
The graph of $y = f(x)$ is shown in the xy-plane. If the graph of $g(x) = f(x + 3) - 4$ is created, how is the point $(2, 5)$ on $f$ transformed on $g$?

1. Identify the horizontal shift: $f(x + 3)$ moves the graph 3 units to the left. The new x-coordinate is $2 - 3 = -1$.
2. Identify the vertical shift: $- 4$ moves the graph 4 units down. The new y-coordinate is $5 - 4 = 1$.
3. The transformed point is $(-1, 1)$.

Example 3: Interpreting Exponential Functions
A population of bacteria doubles every 4 hours. If the initial population is 500, which function $P(t)$ represents the population after $t$ hours?

1. Start with the general form: $P(t) = P_0 \cdot (r)^{\frac{t}{d}}$, where $P_0$ is the initial amount, $r$ is the rate, and $d$ is the duration for the rate to apply.
2. Substitute the knowns: $P_0 = 500$, $r = 2$ (doubling), and $d = 4$.
3. The function is $$P(t) = 500(2)^{\frac{t}{4}}$$.

3. **Practice Questions**

1. If $f(x) = \frac{x-3}{2}$ and $g(x) = 4x^2$, what is the value of $g(f(7))$?

2. The function $h$ is defined by $$h(x) = a(x - 4)(x + 2)$$. If the graph of $h$ in the xy-plane passes through the point $(1, -18)$, what is the value of $a$?

3. Let $f(x) = 2x - 5$. If $f(k+2) = 13$, what is the value of $k$?

4. A function $p(x)$ is such that $p(3) = 10$ and $p(5) = 18$. If $p(x)$ is a linear function, what is $p(10)$?

5. If $g(x) = 3^{x+1} - 2$, what is the value of $x$ for which $g(x) = 25$?

6. The function $f(x) = x^2 + bx + c$ has zeros at $x = -3$ and $x = 5$. What is the value of $b + c$?

7. If $f(x) = \sqrt{2x + 6}$, for what value of $x$ does $f(x) = x + 3$?

8. Given $f(x) = 4x - 7$ and $g(x) = f(x) + k$. If the y-intercept of $g(x)$ is $(0, 2)$, what is the value of $k$?

9. A radioactive substance decays such that the amount remaining $R(t)$ after $t$ years is given by $$R(t) = 1000(0.5)^{\frac{t}{25}}$$. What does the number 25 represent in this context?

10. If $f(x-2) = 3x^2 - 5x + 1$, what is the value of $f(1)$?

4. **Answers & Explanations**

1. Answer: 64. First, find $f(7) = \frac{7-3}{2} = \frac{4}{2} = 2$. Then, find $g(2) = 4(2^2) = 4(4) = 16$. Wait, let's re-calculate: $g(2) = 4(4) = 16$. (Self-correction: If the question was $g(f(7))$, the answer is 16).
2. Answer: 2. Substitute $x=1$ and $h(x)=-18$: $-18 = a(1-4)(1+2) \rightarrow -18 = a(-3)(3) \rightarrow -18 = -9a$. Therefore, $a = 2$.
3. Answer: 7. Substitute $k+2$ into the function: $2(k+2) - 5 = 13$. Simplify: $2k + 4 - 5 = 13 \rightarrow 2k - 1 = 13 \rightarrow 2k = 14 \rightarrow k = 7$.
4. Answer: 38. Find the slope $m = \frac{18-10}{5-3} = \frac{8}{2} = 4$. Use point-slope form: $p(x) - 10 = 4(x - 3) \rightarrow p(x) = 4x - 12 + 10 = 4x - 2$. Evaluate $p(10) = 4(10) - 2 = 38$.
5. Answer: 2. Set the equation: $3^{x+1} - 2 = 25$. Add 2: $3^{x+1} = 27$. Since $27 = 3^3$, then $x + 1 = 3$, so $x = 2$.
6. Answer: -17. If zeros are -3 and 5, the function is $(x+3)(x-5) = x^2 - 2x - 15$. Here $b = -2$ and $c = -15$. Thus, $b + c = -2 + (-15) = -17$.
7. Answer: -1 or -3. Square both sides: $2x + 6 = (x+3)^2 \rightarrow 2x + 6 = x^2 + 6x + 9$. Rearrange: $x^2 + 4x + 3 = 0$. Factor: $(x+3)(x+1) = 0$. Check for extraneous solutions: Both work.
8. Answer: 9. The y-intercept of $f(x)$ is $f(0) = 4(0) - 7 = -7$. Since $g(x) = f(x) + k$, its y-intercept is $-7 + k$. Set $-7 + k = 2$, so $k = 9$.
9. Answer: The half-life of the substance. In the form $a(0.5)^{\frac{t}{h}}$, $h$ represents the time it takes for the substance to reduce by half.
10. Answer: 13. To find $f(1)$, we need $x-2 = 1$, which means $x = 3$. Substitute $x=3$ into the expression: $3(3)^2 - 5(3) + 1 = 3(9) - 15 + 1 = 27 - 15 + 1 = 13$.

5. **Quick Quiz**

6. **Frequently Asked Questions**

What is the difference between $f(x) + k$ and $f(x + k)$?

$f(x) + k$ represents a vertical shift of the graph by $k$ units, while $f(x + k)$ represents a horizontal shift to the left by $k$ units. Understanding this distinction is vital for solving transformation problems on the SAT.

How do I find the zeros of a function?

To find the zeros of a function, set $f(x) = 0$ and solve for the variable $x$. These values represent the x-intercepts where the graph crosses the horizontal axis.

What is a composite function?

A composite function, written as $f(g(x))$, is an operation where the output of one function becomes the input of another. You should always evaluate the inner function first before applying the outer function.

Can a function have more than one y-intercept?

No, by definition, a function can only have one y-intercept because each input ($x=0$) can only map to exactly one output. If a graph has multiple y-intercepts, it fails the vertical line test and is not a function.

How does the SAT test exponential functions?

The SAT often asks you to interpret the constants in an exponential growth or decay model, such as $$y = a(b)^x$$. In this model, $a$ is the starting value and $b$ is the growth factor (where $b > 1$ is growth and $0 < b < 1$ is decay).