At each point, determine the combined function 𝑔 ∘ ℎ and present it as neatly as possible
in terms of. g(x) = x^2-4 h(z) = (2z) / (2-z)

To determine the combined function \( g \circ h \), we need to find \( (g \circ h)(z) \), which means applying the function \( h(z) \) inside the function \( g(x) \). 

Given:
- \( g(x) = x^2 - 4 \)
- \( h(z) = \frac{2z}{2 - z} \)

The composition \( (g \circ h)(z) \) is calculated as follows:

\[
(g \circ h)(z) = g(h(z)) = g\left( \frac{2z}{2 - z} \right)
\]

Now, substitute \( h(z) = \frac{2z}{2 - z} \) into \( g(x) \):

\[
g\left( \frac{2z}{2 - z} \right) = \left( \frac{2z}{2 - z} \right)^2 - 4
\]

Let's simplify step by step:

1. Square the expression:

\[
\left( \frac{2z}{2 - z} \right)^2 = \frac{(2z)^2}{(2 - z)^2} = \frac{4z^2}{(2 - z)^2}
\]

2. Subtract 4:

\[
g(h(z)) = \frac{4z^2}{(2 - z)^2} - 4
\]

3. To subtract 4, express 4 as a fraction with the same denominator:

\[
4 = \frac{4(2 - z)^2}{(2 - z)^2}
\]

Now, subtract:

\[
g(h(z)) = \frac{4z^2}{(2 - z)^2} - \frac{4(2 - z)^2}{(2 - z)^2}
\]

\[
g(h(z)) = \frac{4z^2 - 4(2 - z)^2}{(2 - z)^2}
\]

4. Expand \( (2 - z)^2 \):

\[
(2 - z)^2 = 4 - 4z + z^2
\]

Now substitute back:

\[
g(h(z)) = \frac{4z^2 - 4(4 - 4z + z^2)}{(2 - z)^2}
\]

5. Simplify the numerator:

\[
g(h(z)) = \frac{4z^2 - 16 + 16z - 4z^2}{(2 - z)^2}
\]

\[
g(h(z)) = \frac{16z - 16}{(2 - z)^2}
\]

6. Factor out the common term in the numerator:

\[
g(h(z)) = \frac{16(z - 1)}{(2 - z)^2}
\]

Thus, the combined function is:

\[
(g \circ h)(z) = \frac{16(z - 1)}{(2 - z)^2}
\]

### Do you need more details or have any questions?

Here are 5 related questions for further exploration:
1. What happens to \( g \circ h(z) \) when \( z = 1 \)?
2. How would you find the domain of \( g \circ h(z) \)?
3. What is the range of \( g \circ h(z) \)?
4. How does \( g \circ h(z) \) behave as \( z \to 2 \)?
5. What is the inverse of the combined function \( g \circ h(z) \)?

**Tip:** Always check for restrictions in the domain when composing functions, especially when denominators are involved!

At each point, determine the combined function 𝑔 ∘ ℎ and present it as neatly as possible in terms of g(x) = x^2 - 4, h(z) = (2z) / (2 - z)

Learn how to compose the functions g(x) = x^2 - 4 and h(z) = (2z) / (2 - z) to find (g ∘ h)(z) with a detailed step-by-step explanation. This example covers key algebraic concepts such as function composition and rational function simplification. Suitable for high school students in Grades 10-12.

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