Given the following functions
f（x） =x+5
g (x) = 2x - 1
h (x) = 2x^2 + 9x - 5

Evaluate the following function:
a.
(f o g)(x)
b.（h/f) (x）

Let's evaluate the given composite and quotient functions one by one:

### a. \( (f \circ g)(x) \) 
This means we need to substitute \( g(x) \) into \( f(x) \).

Given:
- \( f(x) = x + 5 \)
- \( g(x) = 2x - 1 \)

To find \( (f \circ g)(x) \), substitute \( g(x) \) into \( f(x) \):

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

Now, apply the expression for \( f(x) \):

\[
f(2x - 1) = (2x - 1) + 5 = 2x + 4
\]

Thus, 
\[
(f \circ g)(x) = 2x + 4
\]

### b. \( \left(\frac{h}{f}\right)(x) \)

This represents the quotient \( \frac{h(x)}{f(x)} \).

Given:
- \( h(x) = 2x^2 + 9x - 5 \)
- \( f(x) = x + 5 \)

To find \( \left(\frac{h}{f}\right)(x) \), divide \( h(x) \) by \( f(x) \):

\[
\left(\frac{h}{f}\right)(x) = \frac{2x^2 + 9x - 5}{x + 5}
\]

We can now perform polynomial long division (or synthetic division) to simplify this expression. Let me proceed with the division.



Given the following functions f(x) = x + 5, g(x) = 2x - 1, h(x) = 2x^2 + 9x - 5, Evaluate the following functions: a. (f o g)(x), b. (h/f)(x)

Learn how to evaluate the composite function (f o g)(x) and the quotient function (h/f)(x) using algebraic methods. This problem involves function composition and polynomial division, suitable for students in Grades 9-12.

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