Question

Attached will be a my written picture of the problem, i understand it’s long.

Answer 1

Use the definitions for the function operations to find the rules of correspondence of the given functions.

`\begin{gathered} f(x)=4x^2+2x+7 \\ g(x)=2x-3 \end{gathered}`

1)

Remember that:

`(g-f)(x)=g(x)-f(x)`

Replace the expressions for g(x) and f(x) and simplify:

`\begin{gathered} (g-f)(x)=(2x-3)-(4x^2+2x+7) \\ =2x-3-4x^2-2x-7 \\ =-4x^2-10 \end{gathered}`

2)

Remember that:

`((f)/(g))(x)=(f(x))/(g(x)),g(x)\\e0`

Replace the expressions for f(x)i and g(x):

`((f)/(g))(x)=(4x^2+2x+7)/(2x-3)`

The domain is the set of all real number such that g(x) is different form 0:

`\begin{gathered} g(x)\\e0 \\ \Rightarrow2x-3\\e0 \\ \Rightarrow2x\\e3 \\ \therefore x\\e(3)/(2) \end{gathered}`

Using interval notation, the domain is:

`(-\infty,(3)/(2))\cup((3)/(2),\infty)`

3)

Remember that:

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

Then:

`\begin{gathered} (f\cdot g)(x)=(4x^2+2x+7)(2x-3) \\ =(4x^2)(2x)+(2x)(2x)+(7)(2x)+(4x^2)(-3)+(2x)(-3)+(7)(-3) \\ =8x^3+4x^2+14x-12x^2-6x-21 \\ =8x^3-8x^2+8x-21 \end{gathered}`

4)

To find f(x-3), replace (x-3) for x in the rule of correspondence of f:

`\begin{gathered} f(x)=4x^2+2x+7 \\ \Rightarrow f(x-3)=4(x-3)^2+2(x-3)+7 \\ =4(x^2-6x+9)+2x-6+7 \\ =4x^2-24x+36+2x+1 \\ =4x^2-22x+37 \end{gathered}`

5)

Remember that:

`(f\circ g)=f(g(x))`

To find f(g(x)), replace g(x) for x into the rule of correspondence of f:

`\begin{gathered} (f\circ g)(x)=f(g(x)) \\ =4(g(x))^2+2(g(x))+7 \end{gathered}`

Replace the expression for g(x):

`\begin{gathered} \Rightarrow(f\circ g)(x)=4(2x-3)^2+2(2x-3)+7 \\ =4(4x^2+-12x+9)+4x-6+7 \\ =16x^2-48x+36+4x+1 \\ =16x^2-44x+37 \end{gathered}`

6)

To find g(f(x)), replace f(x) for x into the rule of correspondence of g(x):

`\begin{gathered} (g\circ f)(x)=g(f(x)) \\ =2\cdot f(x)-3 \\ =2(4x^2+2x+7)-3 \\ =8x^2+4x+14-3 \\ =8x^2+4x+11 \end{gathered}`

7)

Notice that we already have a rule of correspondence for g(f(x)). Substitute x=-1 to find g(f(-1)):

`\begin{gathered} g(f(x))=8x^2+4x+11 \\ \Rightarrow g(f(-1))=8(-1)^2+4(-1)+11 \\ =8-4+11 \\ =15 \end{gathered}`

8)

To find the inverse of f(x), repace y=f(x) and isolate x:

`\begin{gathered} y=4x^2+2x+7 \\ \Rightarrow4x^2+2x+7-y=0 \\ \Rightarrow x=\frac{-2+\sqrt[]{2^2-4(4)(7-y)}}{2(4)} \\ =\frac{-2+\sqrt[]{4-112+16y}}{8} \\ =\frac{-2+\sqrt[]{16y-108}}{8} \\ =\frac{-1+\sqrt[]{4y-27}}{4} \end{gathered}`

Next, switch x and y in the equation:

`y=\frac{-1+\sqrt[]{4x-27}}{4}`

Finally, substitute y=f^-1(x):

`\therefore f^(-1)(x)=\frac{-1+\sqrt[]{4x-27}}{4}`

9)

To find f(-x), replace x for -x in the rule of correspondence of f:

`\begin{gathered} f(-x)=4(-x)^2+2(-x)+7 \\ =4x^2-2x+7 \end{gathered}`