Question

Find the indicated function given f(x)=3x^2-1 and g(x)=2x+4.Simplified answers in descending power of x.

Answer 1

Given the question in the question tab, the following solution steps answer the questions

Step 1: Write the given functions

`\begin{gathered} f(x)=3x^2-1 \\ g(x)=2x+4 \end{gathered}`

Step 2: Find f(g(2))

`\begin{gathered} f(g(2))\text{ means that we first find the inner bracket before finding the outer bracket} \\ \text{That is, we find g(2) first and then find the f(x) of the solution} \\ g(x)=2x+4 \\ g(2)=2(2)+4=4+4=8 \\ f(g(2))=f(8) \\ f(x)=3x^2-1=3(8^2)-1=3(64)-1=192-1=191 \end{gathered}`

The composite function f(g(2)) will give 191

Step 3: Find f(g(x))

`\begin{gathered} f(g(x))\text{ means that we first find the inner bracket before finding the outer bracket} \\ \text{That is, we find g(x) first and then find the f(x) of the solution} \\ g(x)=2x+4 \\ f(x)=3x^2-1 \\ f(g(x))=f(2x+4)=3(2x+4)^2-1 \\ 3(2x+4)^2-1=3(4x^2+16x+16)-1=12x^2+48x+48-1 \\ f(g(x))=12x^2+48x+47_{} \end{gathered}`

Step 4: Find g(f(x))

`\begin{gathered} \text{we find f(x) first and then find the g(x) of the solution} \\ f(x)=3x^2-1 \\ g(x)=2x+4 \\ g(f(x))=g(3x^2-1)=2(3x^2-1)+4 \\ g(f(x))=6x^2-2+4 \\ g(f(x))=6x^2+2 \end{gathered}`

Step 5: Find (gog)(x)

`\begin{gathered} (g\circ g)(x)\Rightarrow g(g(x)) \\ g(x)=2x+4 \\ (g\circ g)(x)=g(2x+4)=2(2x+4)+4=4x+8+4 \\ (g\circ g)(x)=4x+12 \end{gathered}`

Step 6: Find (fof)(-1)

`\begin{gathered} (f\circ f)(x)=f(f(x)) \\ f\mleft(x\mright)=3x^2-1 \\ (f\circ f)(-1)=f(f(-1)) \\ f(-1)=3(-1^2)-1=3(1)-1=3-1=2 \\ f(2)=3(2)^2-1=3(4)-1=12-1 \\ (f\circ f)(-1)=11 \end{gathered}`