Question

Suppose that the functions r and s are defined for all real numbers x as follows. r(x) = 4x ^ 2 s(x) = 2xWrite the expressions for (s + r)(x) and (sr)(x) and evaluate (s - r)(- 1)

Answer 1

a.
`\((s + r)(x) = 2x + 4x^2\`).

b.
`\((s * r)(x) = 8x^3\)`.

c.
`\((s - r)(-1) = -6\)`.

Let's evaluate the expressions for (s + r)(x), (s * r)(x), and (s - r)(-1):

a. (s + r)(x):

`\[(s + r)(x) = s(x) + r(x) = 2x + 4x^2.\]`

b. (s * r)(x):

`\[(s * r)(x) = s(x) \cdot r(x) = (2x) \cdot (4x^2) = 8x^3.\]`

c. (s - r)(-1):

`\[(s - r)(x) = s(x) - r(x) = 2x - 4x^2.\]`

Now, evaluating at x = -1:

`\[(s - r)(-1) = 2(-1) - 4(-1)^2 = -2 - 4 = -6.\]`

The summary is:

a. The expression for
`\((s + r)(x)\) is \(2x + 4x^2\)`.

b. The expression for
`\((s * r)(x)\) is \(8x^3\)`.

c. The value of
`\((s - r)(-1)\) is \(-6\)` .

Answer 2

You have the following expressions:

`\begin{gathered} r(x)=4x^2 \\ s(x)=2x \end{gathered}`

For the required operations between the previous functions, you obtain:

`(s+r)(x)=s(x)+r(x)=4x^2+2x`

The result is 4x^2 + 2x

`(s*r)(x)=s(x)r(x)=4x^2*2x=8x^3`

The result is 8x^3

`\begin{gathered} (s-r)(-1)=s(-1)-r(-1) \\ (s-r)(-1)=4(-1)^2-2(-1)=4+2=6 \end{gathered}`

The result is 6