Question

$$\begin{gathered} \mathrm{For}\mathrm{functions}f\left(x\right)=x^2+1,\mathrm{and}g\left(x\right)=3x-5,\mathrm{find} \\ \left(A\right)(f\circ g)⎠(-1), \\ \left(B\right)(g\circ f)\left(2\right),\mathrm{and} \\ \left(C\right)(f\circ f)(-1). \end{gathered}$$

Short answer

Part(a) 63

Part(b) -14

Part(c) 5

Step-by-step solution

Step 1. Given information

Find

$$\begin{gathered} f\left(x\right)=x^2+1,\mathrm{and}g\left(x\right)=3x-5,\mathrm{find} \\ \left(A\right)(f\circ g)⎠(-1), \\ \left(B\right)(g\circ f)\left(2\right),\mathrm{and} \\ \left(C\right)(f\circ f)(-1). \end{gathered}$$

Part(a)Step 2. Calculation 

$\begin{array}{ll}\text{Use the definition of}(f\circ g)(-1).& (f\circ g)(-1)=f\left(g\right(-1\left)\right)\\ \text{Find}g(-1)\text{where}g\left(x\right)=3x-5.& (f\circ g)(-1)=f(3·(-1)-5)\\ \text{Simplify.}& (f\circ g)(-1)=f(-8)\\ \text{Find}f(-8)\text{where}f\left(x\right)=x^2+1.& (f\circ g)(-1)=(-8{)}^2-1\\ \text{Simplify.}& (f\circ g)(-1)=63\end{array}$

Part (b) Step 3. Calculation 

$\begin{array}{ll}\text{Use the definition of}(g\circ f)\left(2\right).& (g\circ f)\left(2\right)=g\left(f\right(2\left)\right)\\ \text{Find}f\left(2\right)\text{where}f\left(x\right)=x^2+1.& (g\circ f)\left(2\right)=g\left((2{)}^2+1\right)\\ \text{Simplify.}& (g\circ f)\left(2\right)=g\left(5\right)\\ \text{Find}g\left(5\right)\text{where}g\left(x\right)=3x-5.& (g\circ f)\left(2\right)=3(-3)-5\\ \text{Simplify.}& (g\circ f)\left(2\right)=-14\end{array}$

Part(c)Step 4. Calculation

$\begin{array}{ll}\text{Use the definition of}(f\circ f)(-1).& (f\circ f)(-1)=f\left(f\right(-1\left)\right)\\ \text{Find}f\left(2\right)\text{where}f\left(x\right)=x^2+1.& (f\circ f)(-1)=f\left({(-1)}^2+1\right)\\ \text{Simplify.}& (f\circ f)(-1)=f\left(2\right)\\ \text{Find}f\left(2\right)\text{where}f\left(x\right)=x^2+1.& (f\circ f)(-1)=2^2+1\\ \text{Simplify.}& (f\circ f)(-1)=5\end{array}$