Problem 5

Explanation

We want to find the composite function $(g \circ f)(x)$.

Understanding Function Composition

The composite function $(g \circ f)(x)$ is defined as:

This means we will substitute the entire function $f(x)$ into the variable $x$ in the function $g(x)$.

Function Substitution

Given:

• $f(x) = x^2 + 4x - 5$
• $g(x) = 2x - 1$

We substitute $f(x)$ into $g(x)$:

Since $g(x) = 2x - 1$, we replace every $x$ in $g(x)$ with $(x^2 + 4x - 5)$:

Simplifying the Result

Now we distribute the multiplication and simplify:

Conclusion

Thus, the composite function is $2x^2 + 8x - 11$.