Let's tackle the two parts of your question one by one.
- 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.
- Finding \( (g \circ f)(x) \) where \( f(x) = x^2 \) and \( g(x) = 3x - 1 \):
To find \( (g \circ f)(x) \), you compute \( g(f(x)) \):
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First, calculate \( f(x) \):
\[ f(x) = x^2 \]
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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 \( (g \circ f)(x) = 3x^2 - 1 \).