Question

Suppose that the functions g and h are defined for all real numbers x as follows.

g(x) = 3x-6
h(x) = 5x
Write the expressions for (g-h)(x) and (g+h)(x) and evaluate (g-h)(-1) ​

Answer 1

See Below.

Explanation:

We are given the two functions:

`\displaystyle g(x) = 3x - 6 \text{ and } h(x) = 5x`

Part A)

Recall that:

`(g\cdot h)(x)=g(x)\cdot h(x)`

Substitute and simplify:

`\displaystyle \begin{aligned} (g\cdot h)(x) & = (3x-6)\cdot(5x) \\ \\ &=5x(3x)-5x(6) \\ \\&=15x^2-30x \end{aligned}`

Part B)

Recall that:

`(g+h)(x)=g(x)+h(x)`

Substitute and simplify:

`\displaystyle \begin{aligned} g(x) + h(x) & = (3x-6) + (5x) \\ \\ & = 8x- 6 \end{aligned}`

Part C)

Recall that:

`\displaystyle (g-h)(x) = g(x) - h(x)`

Hence:

`\displaystyle \begin{aligned} (g-h)(-1) & = g(-1) - h(-1) \\ \\ & = (3(-1)-6) - (5(-1)) \\ \\ & = (-9) + (5) \\ \\ & = -4\end{aligned}`