Question

1. let (j(x)=sqrt{x + 1}). find (j^{-1}). 7. let (g(x)=2x^{3}-5). write an equation for (h(x)), the inverse of (g). 8. 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}). 9. 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:

6.

Step1: Let \(y = j(x)=\sqrt{x + 1}\)

Swap \(x\) and \(y\): \(x=\sqrt{y + 1}\)

Step2: Solve for \(y\)

Square both sides: \(x^{2}=y + 1\)
Then \(y=x^{2}-1\). Since the original function \(j(x)=\sqrt{x + 1}\) has a domain \(x\geq - 1\) and range \(y\geq0\), the inverse function \(j^{-1}(x)=x^{2}-1,x\geq0\)

Step1: Let \(y = g(x)=2x^{3}-5\)

Swap \(x\) and \(y\): \(x = 2y^{3}-5\)

Step2: Solve for \(y\)

First, add 5 to both sides: \(x + 5=2y^{3}\)
Then divide by 2: \(\frac{x + 5}{2}=y^{3}\)
Take the cube - root of both sides: \(y=\sqrt[3]{\frac{x + 5}{2}}\)
So \(h(x)=g^{-1}(x)=\sqrt[3]{\frac{x + 5}{2}}\)

Step1: Recall the concept of inverse functions

For a function \(y = f(x)\) and its inverse \(x = f^{-1}(y)\), in the inverse function \(f^{-1}\), the independent variable of \(f^{-1}\) is the dependent variable of \(f\), and the dependent variable of \(f^{-1}\) is the independent variable of \(f\).
a. The independent variable of \(f^{-1}\) is the number of calories burned \(c\).
b. The dependent variable of \(f^{-1}\) is the number of hours \(t\) playing soccer.

Step2: Find the inverse function

Since \(c = f(t)\) represents the number of calories burned after playing soccer for \(t\) hours, to find \(f^{-1}\), we need to express \(t\) in terms of \(c\). But we are not given the explicit form of \(f(t)\), so we just use the general concept of inverse - function construction. If we assume we can solve for \(t\) from \(c=f(t)\) to get \(t = f^{-1}(c)\)

Answer:

\(j^{-1}(x)=x^{2}-1,x\geq0\)

7.