Question

find the inverse of g(x)=-3

if f(x)=x2^ and g(x)=3x-1 find (g*f)(x)

Answer 1

Let's tackle the two parts of your question one by one.

1. **Finding the Inverse of \( g(x) = -3 \)**:

The function \( g(x) = -3 \) is a constant function. A constant function does not have an inverse because it does not produce unique outputs for unique inputs; it always outputs \(-3\) regardless of the value of \( x \). Therefore, \( g(x) \) does not have an inverse.

2. **Finding \( (g \circ f)(x) \)** where \( f(x) = x^2 \) and \( g(x) = 3x - 1 \):

To find \( (g \circ f)(x) \), you compute \( g(f(x)) \):

1. First, calculate \( f(x) \):

\[
f(x) = x^2
\]

2. Next, substitute \( f(x) \) into \( g(x) \):

\[
g(f(x)) = g(x^2) = 3(x^2) - 1
\]

This simplifies to:

\[
g(f(x)) = 3x^2 - 1
\]

So, the composition \( (g \circ f)(x) = 3x^2 - 1 \).

### Summary:
- The inverse of \( g(x) = -3 \) does not exist.
- The composition \( (g \circ f)(x) = 3x^2 - 1 \).