Manish writes the functions g(x) = RootIndex 3 StartRoot negative x EndRoot – 72 and h(x) = –(x + 72)3.

Which pair of expressions could Manish use to show that g(x) and h(x) are inverse functions?

To determine if \( g(x) \) and \( h(x) \) are inverse functions, we must check if applying one function after the other returns the original input value \( x \). In other words, if \( g(x) \) and \( h(x) \) are inverses, then:

1. \( g(h(x)) = x \)
2. \( h(g(x)) = x \)

Let's examine both \( g(x) \) and \( h(x) \):

Given:
\[
g(x) = \sqrt[3]{-x} - 72
\]
and
\[
h(x) = -(x + 72)^3
\]

### Step 1: Finding \( g(h(x)) \)
Substitute \( h(x) = -(x + 72)^3 \) into \( g(x) \):
\[
g(h(x)) = g(-(x + 72)^3)
\]
Using \( g(x) = \sqrt[3]{-x} - 72 \):
\[
g(h(x)) = \sqrt[3]{-(-(x + 72)^3)} - 72
\]
\[
= \sqrt[3]{(x + 72)^3} - 72
\]
Since the cube root of \( (x + 72)^3 \) is \( x + 72 \), we get:
\[
g(h(x)) = (x + 72) - 72 = x
\]

### Step 2: Finding \( h(g(x)) \)
Substitute \( g(x) = \sqrt[3]{-x} - 72 \) into \( h(x) \):
\[
h(g(x)) = h\left(\sqrt[3]{-x} - 72\right)
\]
Using \( h(x) = -(x + 72)^3 \):
\[
h(g(x)) = -\left(\left(\sqrt[3]{-x} - 72\right) + 72\right)^3
\]
\[
= -\left(\sqrt[3]{-x}\right)^3
\]
Since the cube of \( \sqrt[3]{-x} \) is \( -x \), we get:
\[
h(g(x)) = -(-x) = x
\]

Since both \( g(h(x)) = x \) and \( h(g(x)) = x \), we confirm that \( g(x) \) and \( h(x) \) are inverse functions.

### Answer
The pair of expressions Manish could use to show that \( g(x) \) and \( h(x) \) are inverses is:
\[
g(h(x)) = x \quad \text{and} \quad h(g(x)) = x
\]

Would you like more details or have questions?

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**Related Questions:**
1. How do we determine if two functions are inverses algebraically?
2. What are common properties of inverse functions?
3. How can we graphically verify that two functions are inverses?
4. How does the notation of inverse functions work?
5. What is the importance of the domain and range in inverse functions?

**Tip:** When checking if two functions are inverses, remember to test both compositions \( f(g(x)) = x \) and \( g(f(x)) = x \); both must equal \( x \) for the functions to be true inverses.

Manish writes the functions g(x) = RootIndex 3 StartRoot negative x EndRoot – 72 and h(x) = –(x + 72)3. Which pair of expressions could Manish use to show that g(x) and h(x) are inverse functions?

Explore how to determine if g(x) = RootIndex 3 StartRoot -x EndRoot – 72 and h(x) = –(x + 72)^3 are inverse functions by using function composition. This problem demonstrates key algebraic concepts like inverse functions and function composition, ideal for students in Grades 10-12.

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