g(x)=x-5, h(x)=3x^2 write the expressions for (g•h)(x), (g-h)(x), and evaluate (g+h)(2)

Given the functions \( g(x) = x - 5 \) and \( h(x) = 3x^2 \), let's find the expressions for \((g \circ h)(x)\), \((g - h)(x)\), and evaluate \((g + h)(2)\).

### 1. \((g \circ h)(x)\)
The composition \((g \circ h)(x)\) means \( g(h(x)) \). 

Substitute \( h(x) = 3x^2 \) into \( g(x) \):

\[
(g \circ h)(x) = g(3x^2) = 3x^2 - 5
\]

So, \((g \circ h)(x) = 3x^2 - 5\).

### 2. \((g - h)(x)\)
This represents the difference between \( g(x) \) and \( h(x) \):

\[
(g - h)(x) = g(x) - h(x) = (x - 5) - 3x^2
\]

So, \((g - h)(x) = -3x^2 + x - 5\).

### 3. \((g + h)(2)\)
This represents the sum of \( g(x) \) and \( h(x) \) evaluated at \( x = 2 \):

First, calculate \( g(2) \) and \( h(2) \):

\[
g(2) = 2 - 5 = -3
\]
\[
h(2) = 3(2)^2 = 3(4) = 12
\]

Now, add the results:

\[
(g + h)(2) = g(2) + h(2) = -3 + 12 = 9
\]

So, \((g + h)(2) = 9\).

Would you like more details or have any other questions?

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Here are 8 related questions to consider:

1. What is \((h \circ g)(x)\)?
2. How do you find the inverse of \(g(x)\) and \(h(x)\)?
3. What is the domain and range of \(g(x)\) and \(h(x)\)?
4. How do you graph \(g(x)\) and \(h(x)\) on the same coordinate plane?
5. What is the derivative of \(g(x)\) and \(h(x)\)?
6. How do you solve the equation \(g(x) = h(x)\)?
7. What happens to \((g \circ h)(x)\) if \(h(x) = x^2\) instead of \(3x^2\)?
8. How do you find the zeros of \((g - h)(x)\)?

**Tip:** When composing functions like \((g \circ h)(x)\), always substitute the inner function into the outer function carefully to avoid mistakes.

Explore how to compose functions and perform arithmetic operations on functions using examples of g(x) = x - 5 and h(x) = 3x^2. Learn about (g • h)(x), (g - h)(x), and evaluate (g + h)(2). Suitable for students in Grades 10-12.

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