Question

For functions $f\left(x\right)=5x^2-1$and

$g\left(x\right)=4x-1$find

a.$f\circ g\left(1\right)$

b.$g\circ f(-1)$

c.$f\circ f\left(2\right)$

Short answer

The value is $44,15,1804$respectively

Step-by-step solution

Given information

Given the function

$$\begin{gathered} f\left(x\right)=5x^2-1 \\ g\left(x\right)=4x-1 \end{gathered}$$

Calculate g(1) by substituting in g(x)

Substituting, $x=1$, we get

role="math" localid="1644935694008" $$\begin{gathered} g\left(1\right)=f(4x-1) \\ g\left(1\right)=f\left(4\right(1)-1) \\ g\left(1\right)=f\left(3\right) \end{gathered}$$

Calculate f(3) by substituting in the function

Substituting , we get

$$\begin{gathered} f\left(3\right)=5x^2-1 \\ f\left(3\right)=5{\left(3\right)}^2-1 \\ f\left(3\right)=44 \end{gathered}$$

Calculate f(-1)  by substituting in the value

Substituting, we get

$$\begin{gathered} f(-1)=5{(-1)}^2-1 \\ f(-1)=5-1 \\ f(-1)=4 \end{gathered}$$

Calculate g(4) by substituting the value in the function

Substituting the values, we get

$$\begin{gathered} g\left(4\right)=4x-1 \\ g\left(4\right)=4\left(4\right)-1 \\ g\left(4\right)=15 \\ g\circ f(-1)=15 \end{gathered}$$

Step 6:Substitute f(2) by substituting in the values

Substituting, we get

$$\begin{gathered} f\left(2\right)=5x^2-1 \\ f\left(2\right)=5{\left(2\right)}^2-1 \\ f\left(2\right)=19 \end{gathered}$$

Calculating f(19) by substituting in the values 

Substituting the values, we get

$$\begin{gathered} f\left(19\right)=5x^2-1 \\ f\left(19\right)=5{\left(19\right)}^2-1 \\ f\left(19\right)=1804 \end{gathered}$$