The image provides the functions $f(x) = x^2 - 4x - 1$ and $g(x) = \frac{9x - 5}{x - 3}$, and you're asked to find $f(x + 7)$ and $g\left(\frac{x}{2}\right)$.

1. Find $f(x + 7)$

To find $f(x + 7)$, we substitute $x + 7$ into the function $f(x)$: $f(x + 7) = (x + 7)^2 - 4(x + 7) - 1$

Now, simplify step-by-step:

$(x + 7)^2 = x^2 + 14x + 49$ $-4(x + 7) = -4x - 28$

Now substitute these into the function: $f(x + 7) = x^2 + 14x + 49 - 4x - 28 - 1$

Combine like terms: $f(x + 7) = x^2 + 10x + 20$

2. Find $g\left(\frac{x}{2}\right)$

For $g\left(\frac{x}{2}\right)$, substitute $\frac{x}{2}$ into the function $g(x)$:

$g\left(\frac{x}{2}\right) = \frac{9\left(\frac{x}{2}\right) - 5}{\frac{x}{2} - 3}$

Simplify the numerator: $9\left(\frac{x}{2}\right) = \frac{9x}{2}$ $\frac{9x}{2} - 5 = \frac{9x - 10}{2}$

Simplify the denominator: $\frac{x}{2} - 3 = \frac{x - 6}{2}$

Now the expression becomes: $g\left(\frac{x}{2}\right) = \frac{\frac{9x - 10}{2}}{\frac{x - 6}{2}}$

Since both the numerator and denominator are divided by 2, they cancel out: $g\left(\frac{x}{2}\right) = \frac{9x - 10}{x - 6}$

Final Answers:

• $f(x + 7) = x^2 + 10x + 20$
• $g\left(\frac{x}{2}\right) = \frac{9x - 10}{x - 6}$

Would you like any more details or explanations?

Here are 5 related questions for practice:

1. What is $f(x - 3)$ for the given function $f(x)$?
2. How would you solve $f(g(x))$ for the functions given?
3. Can you find the inverse of the function $g(x)$?
4. What happens if you solve $f(2x)$ instead of $f(x + 7)$?
5. How would you graph both $f(x)$ and $g(x)$ on the same coordinate plane?

Tip: Always expand and simplify step-by-step when substituting expressions into functions to avoid mistakes.