Question

1. let (y(x)=sqrt{x + 1}). find (y^{-1}).
2. let (g(x)=2x^{3}-5). write an equation for (h(x)), the inverse of (g).
3. consider the function (h = f(g)=sqrt{2g - 1}).

a. what is the independent variable of (f^{-1})?
b. what is the dependent variable of (f^{-1}).
c. find (f^{-1}).

1. let (c = f(t)) represent the number of calories burned after playing soccer for (t) hours. interpret the meaning of the function (t = f^{-1}(c)).

Explanation:

Step1: Recall inverse - function concept

For a function \(y = f(x)\), to find its inverse, we swap \(x\) and \(y\) and solve for \(y\).

Step2: Solve for problem 6

Let \(y = \sqrt{x}+1\). First, swap \(x\) and \(y\): \(x=\sqrt{y}+1\). Then solve for \(y\):
\[

$$\begin{align*} x - 1&=\sqrt{y}\\ y&=(x - 1)^2,x\geq1 \end{align*}$$

\]
So \(y^{-1}=(x - 1)^2,x\geq1\)

Step3: Solve for problem 7

Let \(y = g(x)=2x^{3}-5\). Swap \(x\) and \(y\): \(x = 2y^{3}-5\). Then solve for \(y\):
\[

$$\begin{align*} x + 5&=2y^{3}\\ y^{3}&=\frac{x + 5}{2}\\ y&=\sqrt[3]{\frac{x + 5}{2}} \end{align*}$$

\]
So \(h(x)=\sqrt[3]{\frac{x + 5}{2}}\)

Step4: Solve for problem 8a

For a function \(h = f(g)=\sqrt{2g-1}\), if \(y = f(g)\), then for the inverse function \(f^{-1}\), the independent - variable of \(f^{-1}\) is the range of \(f\). The independent variable of \(f^{-1}\) is \(h\).

Step5: Solve for problem 8b

The dependent variable of \(f^{-1}\) is \(g\).

Step6: Solve for problem 8c

Let \(h=\sqrt{2g - 1}\). Swap \(h\) and \(g\): \(g=\sqrt{2h-1}\). Then solve for \(h\):
\[

$$\begin{align*} g^{2}&=2h-1\\ 2h&=g^{2}+1\\ h&=\frac{g^{2}+1}{2},g\geq0 \end{align*}$$

\]
So \(f^{-1}(g)=\frac{g^{2}+1}{2},g\geq0\)

Step7: Solve for problem 9

The function \(c = f(t)\) represents the number of calories burned after playing soccer for \(t\) hours. The function \(t = f^{-1}(c)\) represents the number of hours of playing soccer to burn \(c\) calories.

Answer:

1. \(y^{-1}=(x - 1)^2,x\geq1\)
2. \(h(x)=\sqrt[3]{\frac{x + 5}{2}}\)
3. a. \(h\)

b. \(g\)
c. \(f^{-1}(g)=\frac{g^{2}+1}{2},g\geq0\)

1. \(t = f^{-1}(c)\) represents the number of hours of playing soccer to burn \(c\) calories.