(b)  Express the height h as a function of weight W.  Verify your answer by checking that 

​W(h(W))equals=W 

and 

​h(W(h))equals=h.

### Part (b): Expressing Height \( h \) as a Function of Weight \( W \)

Given the original function:
\[
W(h) = 50 + 2.4(h - 61)
\]
we want to solve for \( h \) in terms of \( W \).

### Step 1: Isolate the term involving \( h \)
\[
W = 50 + 2.4(h - 61)
\]
Subtract 50 from both sides:
\[
W - 50 = 2.4(h - 61)
\]

### Step 2: Solve for \( h \)
Divide both sides by 2.4:
\[
\frac{W - 50}{2.4} = h - 61
\]
Add 61 to both sides to solve for \( h \):
\[
h = \frac{W - 50}{2.4} + 61
\]

So, the height \( h \) as a function of weight \( W \) is:
\[
h(W) = \frac{W - 50}{2.4} + 61
\]

### Step 3: Verification
We need to verify two things:

1. \( W(h(W)) = W \)
2. \( h(W(h)) = h \)

#### Verify \( W(h(W)) = W \)
Start by substituting \( h(W) \) into the original function:
\[
W(h(W)) = 50 + 2.4\left(\left(\frac{W - 50}{2.4} + 61\right) - 61\right)
\]
Simplify inside the parentheses:
\[
W(h(W)) = 50 + 2.4\left(\frac{W - 50}{2.4}\right)
\]
The 2.4's cancel out:
\[
W(h(W)) = 50 + (W - 50)
\]
Simplify:
\[
W(h(W)) = W
\]
This verifies that \( W(h(W)) = W \).

#### Verify \( h(W(h)) = h \)
Start by substituting \( W(h) \) into the inverse function:
\[
h(W(h)) = \frac{W(h) - 50}{2.4} + 61
\]
Substitute \( W(h) = 50 + 2.4(h - 61) \):
\[
h(W(h)) = \frac{50 + 2.4(h - 61) - 50}{2.4} + 61
\]
Simplify inside the fraction:
\[
h(W(h)) = \frac{2.4(h - 61)}{2.4} + 61
\]
The 2.4's cancel out:
\[
h(W(h)) = h - 61 + 61
\]
Simplify:
\[
h(W(h)) = h
\]
This verifies that \( h(W(h)) = h \).

Both verifications confirm that the expressions for \( h(W) \) and \( W(h) \) are correct.

Would you like more information or have any questions? 

Here are some related questions for further exploration:

1. How would you interpret the inverse function \( h(W) \) in a real-world context?
2. Can you derive the inverse function for a different linear equation?
3. How would the formula for \( h(W) \) change if the coefficient 2.4 were different?
4. What is the significance of verifying \( W(h(W)) = W \) and \( h(W(h)) = h \)?
5. Can the process of finding an inverse function be applied to a non-linear equation?

**Tip:** When working with inverse functions, always substitute back to verify that the original and inverse functions are correctly related.

Learn how to express height h as a function of weight W with a detailed step-by-step solution. This problem covers linear equations and inverse functions, verifying the relationship between height and weight mathematically.

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