
Composite functions g(x)=x+1/x and h(z)=1/2z+1

Given two functions \( g(x) = x + \frac{1}{x} \) and \( h(z) = \frac{1}{2z} + 1 \), we can evaluate their composition in various ways.

### 1. Composite Function \( g(h(z)) \)
To find \( g(h(z)) \), we substitute \( h(z) \) into \( g(x) \):

\[
g(h(z)) = g\left( \frac{1}{2z} + 1 \right) = \left( \frac{1}{2z} + 1 \right) + \frac{1}{\frac{1}{2z} + 1}
\]

Now simplify this expression:

\[
g(h(z)) = \frac{1}{2z} + 1 + \frac{1}{\frac{1}{2z} + 1}
\]

To simplify \( \frac{1}{\frac{1}{2z} + 1} \), we can combine the terms in the denominator:

\[
\frac{1}{\frac{1}{2z} + 1} = \frac{1}{\frac{1 + 2z}{2z}} = \frac{2z}{1 + 2z}
\]

Thus,

\[
g(h(z)) = \frac{1}{2z} + 1 + \frac{2z}{1 + 2z}
\]

### 2. Composite Function \( h(g(x)) \)
Now, to find \( h(g(x)) \), we substitute \( g(x) \) into \( h(z) \):

\[
h(g(x)) = h\left( x + \frac{1}{x} \right) = \frac{1}{2\left( x + \frac{1}{x} \right)} + 1
\]

Simplify the expression:

\[
h(g(x)) = \frac{1}{2x + \frac{2}{x}} + 1 = \frac{1}{\frac{2x^2 + 2}{x}} + 1 = \frac{x}{2x^2 + 2} + 1
\]

Thus,

\[
h(g(x)) = \frac{x}{2(x^2 + 1)} + 1
\]

### Summary
- \( g(h(z)) = \frac{1}{2z} + 1 + \frac{2z}{1 + 2z} \)
- \( h(g(x)) = \frac{x}{2(x^2 + 1)} + 1 \)

Would you like further clarification on any of the steps?

Here are 5 related questions to explore:
1. How can we verify the domain of the composite function \( g(h(z)) \)?
2. What happens to \( g(h(z)) \) when \( z = 1 \)?
3. How do you find the inverse of the function \( g(x) \)?
4. Can composite functions like \( g(h(z)) \) have real-world applications?
5. How can you graph the functions \( g(x) \) and \( h(z) \)?

**Tip**: Always check the domains of functions before composing them, as certain inputs may not be valid after composition.

Composite functions g(x)=x+1/x and h(z)=1/2z+1

Learn how to evaluate composite functions such as g(x)=x + 1/x and h(z)=1/(2z) + 1. This step-by-step guide explains how to simplify these functions and explore their composition, ideal for Grades 10-12.

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