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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.

Find f(x + 7) and g(x/2) where f(x) = x^2 - 4x - 1 and g(x) = (9x - 5)/(x - 3). Simplify the answers as much as possible.

This tutorial provides a detailed step-by-step solution for finding f(x + 7) and g(x/2) from the given functions f(x) = x^2 - 4x - 1 and g(x) = (9x - 5)/(x - 3). Key algebraic concepts like function substitution, quadratic expansion, and rational function simplification are covered. Suitable for students in Grades 10-12.

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