Answer to Question #272427 in Calculus for Yangyang

Question

For each of the following functions determine the inverse image of T = {x ∈ R : 0 ≤ x 2 − 25}.

1. f : R → R defined by f(x) = 3x3.

2. g : R + → R defined by g(x) = ln(x).

3. h : R → R defined by h(x) = x − 9.

Answer 1

1.

a. Write $y=f(x)$

$$y = 3x^3.$$

b. Interchange $x$ and $y$

$$x=3y^3$$

c. Solve for $y$

$$y=(x/3)^{1/3}$$

d. Write $f^{-1}(x)=y$

$$f^{-1}(x)=(x/3)^{1/3}$$

$$0\leq x^2\leq25=>-5\leq x\leq5$$

$$f^{-1}(T)=[-(5/3)^{1/3}, (5/3)^{1/3}]$$

2.

a. Write $y=g(x)$

$$y = \ln x$$

b. Interchange $x$ and $y$

$$x=\ln y$$

c. Solve for $y$

$$y=e^x$$

d. Write $g^{-1}(x)=y$

$$g^{-1}(x)=e^x,x>0, x\in \R$$

$$0\leq x^2\leq25, x>0=>0<x\leq5$$

$$g^{-1}(T)=[1, e^5]$$

3.

a. Write $y=h(x)$

$$y = x-9.$$

b. Interchange $x$ and $y$

$$x=y-9$$

c. Solve for $y$

$$y=x+9$$

d. Write $h^{-1}(x)=y$

$$h^{-1}(x)=x+9$$

$$0\leq x^2\leq25=>-5\leq x\leq5$$

$$h^{-1}(T)=[4, 14]$$