Question

example 4

1. weight the formula to convert weight in pounds to stones is $p(x) = \frac{x}{14}$, where $x$ is the weight in pounds.

a. find the inverse of $p(x)$, and describe its meaning.
b. graph $p(x)$ and $p^{-1}(x)$. use the graph to find the weight in pounds of a dog that weighs about 2.5 stones.
lesson 6-2 • inverse r

Explanation:

Part (a)

Step 1: Replace \( p(x) \) with \( y \)

We start with the function \( p(x)=\frac{x}{14} \). Replace \( p(x) \) with \( y \), so we have \( y = \frac{x}{14} \).

Step 2: Swap \( x \) and \( y \)

To find the inverse, we swap the roles of \( x \) and \( y \). This gives us \( x=\frac{y}{14} \).

Step 3: Solve for \( y \)

Multiply both sides of the equation \( x = \frac{y}{14} \) by 14 to isolate \( y \). So, \( y=14x \).

Step 4: Replace \( y \) with \( p^{-1}(x) \)

The inverse function is \( p^{-1}(x) = 14x \). The meaning of the inverse function is that it converts weight in stones to weight in pounds. If \( x \) is the weight in stones, then \( p^{-1}(x) \) gives the weight in pounds.

Part (b)

Step 1: Analyze the function \( p(x)=\frac{x}{14} \)

The function \( p(x)=\frac{x}{14} \) is a linear function with a slope of \( \frac{1}{14} \) and a \( y \)-intercept at \( (0,0) \). We can plot some points, for example, when \( x = 0 \), \( p(0)=0 \); when \( x = 14 \), \( p(14) = 1 \).

Step 2: Analyze the inverse function \( p^{-1}(x)=14x \)

The inverse function \( p^{-1}(x)=14x \) is also a linear function with a slope of 14 and a \( y \)-intercept at \( (0,0) \). We can plot some points, for example, when \( x = 0 \), \( p^{-1}(0)=0 \); when \( x = 1 \), \( p^{-1}(1)=14 \).

Step 3: Use the graph to find the weight in pounds for 2.5 stones

We know that the inverse function \( p^{-1}(x) \) converts stones to pounds. So, we substitute \( x = 2.5 \) into \( p^{-1}(x) \). Using the inverse function \( p^{-1}(x)=14x \), we have \( p^{-1}(2.5)=14\times2.5 = 35 \). So, a dog that weighs 2.5 stones weighs 35 pounds.

Answer:

s:
a. The inverse function is \( \boldsymbol{p^{-1}(x)=14x} \), and it converts weight in stones to weight in pounds.
b. The weight of the dog in pounds is \( \boldsymbol{35} \).