https://file.aiquickdraw.com/mathbot/file/1727481736484mrtn7h50.jpg

The image shows two functions \( f(x) \) and \( g(x) \), defined as follows:

\[
f(x) = \frac{4x + 3}{5x + 2}, \quad g(x) = \sqrt{x^2 - 8x}
\]

You are asked to find:

1. \( f\left( \frac{5}{x} \right) \)
2. \( g(x - 3) \)

### 1. Finding \( f\left( \frac{5}{x} \right) \):

We substitute \( \frac{5}{x} \) into the function \( f(x) \):

\[
f\left( \frac{5}{x} \right) = \frac{4 \cdot \frac{5}{x} + 3}{5 \cdot \frac{5}{x} + 2}
= \frac{\frac{20}{x} + 3}{\frac{25}{x} + 2}
\]

Now, simplify both the numerator and the denominator:

\[
= \frac{\frac{20 + 3x}{x}}{\frac{25 + 2x}{x}} = \frac{20 + 3x}{25 + 2x}
\]

### 2. Finding \( g(x - 3) \):

We substitute \( x - 3 \) into the function \( g(x) \):

\[
g(x - 3) = \sqrt{(x - 3)^2 - 8(x - 3)}
\]

First, expand \( (x - 3)^2 \) and simplify:

\[
(x - 3)^2 = x^2 - 6x + 9
\]

Then:

\[
g(x - 3) = \sqrt{x^2 - 6x + 9 - 8(x - 3)}
\]
\[
g(x - 3) = \sqrt{x^2 - 6x + 9 - 8x + 24} = \sqrt{x^2 - 14x + 33}
\]

### Final Answers:

1. \( f\left( \frac{5}{x} \right) = \frac{20 + 3x}{25 + 2x} \)
2. \( g(x - 3) = \sqrt{x^2 - 14x + 33} \)

Would you like a more detailed explanation on any step?

Here are five related questions to deepen your understanding:

1. How do you handle functions with fractions when substituting a complex term?
2. What strategies can you use to simplify square root expressions like \( g(x - 3) \)?
3. How would the function \( f(x) \) change if the coefficients in the numerator or denominator were different?
4. Can you graph the functions \( f(x) \) and \( g(x) \) based on these transformations?
5. How do substitutions in functions affect their domain and range?

**Tip:** When simplifying expressions, always check if the result can be further reduced or factored to its simplest form.

Find f(5/x) and g(x - 3) where f(x) = (4x + 3) / (5x + 2) and g(x) = √(x² - 8x).

Learn how to substitute and simplify functions f(x) and g(x) where f(x) = (4x + 3) / (5x + 2) and g(x) = √(x² - 8x). This problem includes algebraic substitution and simplification techniques. Suitable for students in Grades 9-11.

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