Four functions are given, f(x)=3x+5, g(x)=x/3  -5, h(x)= x/3  -  5/3, j(x)= 3x+15    solve f(g(x)) and g(f(x))

Let's solve for both \( f(g(x)) \) and \( g(f(x)) \) step by step.

### 1. Solving \( f(g(x)) \):
The function \( f(x) = 3x + 5 \) and \( g(x) = \frac{x}{3} - 5 \).

To find \( f(g(x)) \), substitute \( g(x) \) into \( f(x) \):

\[
f(g(x)) = f\left( \frac{x}{3} - 5 \right)
\]

Now apply the function \( f(x) = 3x + 5 \):

\[
f(g(x)) = 3\left( \frac{x}{3} - 5 \right) + 5
\]

Distribute the 3:

\[
f(g(x)) = 3 \times \frac{x}{3} - 3 \times 5 + 5 = x - 15 + 5
\]

Simplify:

\[
f(g(x)) = x - 10
\]

### 2. Solving \( g(f(x)) \):
Now, we want to find \( g(f(x)) \), where \( g(x) = \frac{x}{3} - 5 \) and \( f(x) = 3x + 5 \).

Substitute \( f(x) \) into \( g(x) \):

\[
g(f(x)) = g(3x + 5)
\]

Now apply the function \( g(x) = \frac{x}{3} - 5 \):

\[
g(f(x)) = \frac{3x + 5}{3} - 5
\]

Simplify:

\[
g(f(x)) = \frac{3x}{3} + \frac{5}{3} - 5 = x + \frac{5}{3} - 5
\]

Combine terms:

\[
g(f(x)) = x + \frac{5}{3} - \frac{15}{3} = x - \frac{10}{3}
\]

### Final Results:
- \( f(g(x)) = x - 10 \)
- \( g(f(x)) = x - \frac{10}{3} \)

Would you like more details on any step or further clarifications?

Here are 5 related questions to explore:
1. How would the result change if \( f(x) \) were a quadratic function?
2. Can you compute \( f(h(x)) \) and \( h(f(x)) \)?
3. What happens if you compose \( f(x) \) with itself, i.e., \( f(f(x)) \)?
4. What are the inverses of \( f(x) \) and \( g(x) \), and can you verify the compositions using them?
5. How would you approach composing three functions, say \( f(g(h(x))) \)?

**Tip:** Function composition can be thought of as feeding the output of one function as the input to another. Understanding the inner workings of each function helps simplify the overall expression.

Four functions are given, f(x)=3x+5, g(x)=x/3 -5, h(x)=x/3 - 5/3, j(x)= 3x+15 solve f(g(x)) and g(f(x))

This detailed explanation walks through solving function compositions f(g(x)) and g(f(x)) where f(x) = 3x + 5 and g(x) = x/3 - 5. Learn how to substitute and simplify linear functions step by step. Suitable for students in Grades 9-11.

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