Question

Let g(x)=x^2-64 and h(x)= (long division symbol)8x+512. Find each combination and state its domain.1. (h/g)(x)(Number one has a fraction)2. (h • g)(x)3. (g • h)(x)Show work please!

Answer 1

The given functions are:

`\begin{gathered} g(x)=x^2-64 \\ h(x)=8x+512 \end{gathered}`

1. (h/g)(x):

`((h)/(g))(x)=(h(x))/(g(x))`

Then we need to divide h(x) by g(x):

`(8x+512)/(x^2-64)`

The factored form of the denominator is:

`\begin{gathered} x^2-64=(x+8)(x-8) \\ \text{Then} \\ (8x+512)/((x+8)(x-8)) \end{gathered}`

We have to use partial fractions by applying the following model:

`(8x+512)/((x+8)(x-8))=(a_0)/((x+8))+(a_1)/((x-8))`

Multiply both sides of the equations by the denominator:

`((8x+512)(x+8)(x-8))/((x+8)(x-8))=(a_0(x+8)(x-8))/((x+8))+(a_1(x+8)(x-8))/((x-8))`

Now, simplify:

`(8x+512)=a_0(x-8)+a_1(x+8)`

To find a0 and a1, substitute the roots of the factors x=-8 and x=8, and solve:

`\begin{gathered} x=8 \\ (8\cdot8+512)=a_0(8-8)+a_1(8+8) \\ 64+512=a_0(0)+a_1(16) \\ 576=a_1(16) \\ a_1=(576)/(16)=36 \\ \text{For x=-8:} \\ (8\cdot-8+512)=a_0(-8-8)+a_1(-8+8) \\ -64+512=a_0(-16)+a_1(0) \\ 448=a_0(-16) \\ a_0=(448)/(-16) \\ a_0=-28 \end{gathered}`

Now, replace these values and find the equation:

`\begin{gathered} (a_0)/((x+8))+(a_1)/((x-8))=\frac{-28_{}}{x+8}+\frac{36_{}}{x-8} \\ ((h)/(g))(x)=\frac{-28_{}}{x+8}+\frac{36_{}}{x-8} \end{gathered}`

2. (h*g)(x):

`\begin{gathered} (h\cdot g)(x)=h(x)\cdot g(x) \\ (h\cdot g)(x)=(8x+512)(x^2-64) \end{gathered}`

Apply the distributive property:

`\begin{gathered} (h\cdot g)(x)=8x\cdot x^2+8x\cdot(-64)+512\cdot x^2+512\cdot(-64) \\ (h\cdot g)(x)=8x^3-512x+512x^2-32768 \\ And\text{ reorder terms:} \\ (h\cdot g)(x)=8x^3+512x^2-512x-32768 \end{gathered}`

3. (g*h)(x):

`\begin{gathered} (g\cdot h)(x)=g(x)\cdot h(x) \\ (g\cdot h)(x)=(x^2-64)\cdot\mleft(8x+512\mright) \end{gathered}`

The result will be the same, but let's solve it to check. Apply the distributive property:

`\begin{gathered} (g\cdot h)(x)=x^2\cdot8x+x^2\cdot512-64\cdot8x-64\cdot512 \\ (g\cdot h)(x)=8x^3+512x^2-512x-32768 \end{gathered}`