Question

Find the inverse of the following functions.
A. g(x)=x³+1
B. t(x)=3x_4
C. h(x)=3square root x+2_2
D. f(x)=x+4÷x_3

Answer 1

Let's find the inverse of each of the functions step by step:

### A. \( g(x) = x^3 + 1 \)

1. Start by replacing \( g(x) \) with \( y \):
\[
y = x^3 + 1
\]
2. Swap \( x \) and \( y \):
\[
x = y^3 + 1
\]
3. Solve for \( y \):
\[
y^3 = x - 1
\]
\[
y = \sqrt[3]{x - 1}
\]
4. The inverse function is:
\[
g^{-1}(x) = \sqrt[3]{x - 1}
\]

### B. \( t(x) = 3x^4 \)

1. Start by replacing \( t(x) \) with \( y \):
\[
y = 3x^4
\]
2. Swap \( x \) and \( y \):
\[
x = 3y^4
\]
3. Solve for \( y \):
\[
y^4 = \frac{x}{3}
\]
\[
y = \sqrt[4]{\frac{x}{3}}
\]
4. The inverse function is:
\[
t^{-1}(x) = \sqrt[4]{\frac{x}{3}}
\]

### C. \( h(x) = 3\sqrt{x} + 2 \)

1. Start by replacing \( h(x) \) with \( y \):
\[
y = 3\sqrt{x} + 2
\]
2. Swap \( x \) and \( y \):
\[
x = 3\sqrt{y} + 2
\]
3. Solve for \( y \):
\[
x - 2 = 3\sqrt{y}
\]
\[
\frac{x - 2}{3} = \sqrt{y}
\]
\[
y = \left(\frac{x - 2}{3}\right)^2
\]
4. The inverse function is:
\[
h^{-1}(x) = \left(\frac{x - 2}{3}\right)^2
\]

### D. \( f(x) = \frac{x + 4}{x} \)

1. Start by replacing \( f(x) \) with \( y \):
\[
y = \frac{x + 4}{x}
\]
2. Multiply both sides by \( x \):
\[
yx = x + 4
\]
3. Rearrange to isolate \( x \):
\[
yx - x = 4
\]
\[
x(y - 1) = 4
\]
\[
x = \frac{4}{y - 1}
\]
4. Swap \( x \) and \( y \):
\[
y = \frac{4}{x - 1}
\]
5. The inverse function is:
\[
f^{-1}(x) = \frac{4}{x - 1}
\]

### Summary of Inverses:
- \( g^{-1}(x) = \sqrt[3]{x - 1} \)
- \( t^{-1}(x) = \sqrt[4]{\frac{x}{3}} \)
- \( h^{-1}(x) = \left(\frac{x - 2}{3}\right)^2 \)
- \( f^{-1}(x) = \frac{4}{x - 1} \)