Question

Let f(x) = 2x - 1 and g(x) = x² + 1. Find and simplify: (g ° ƒ) (1/2)

Answer 1

To find the composition we need, let's first find the general expression:

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

Hence:

$(g\circ f)(x)=4x^2-4x+2$

Now we can evaluate when x=1/2:

$\begin{gathered} (g\circ f)(\frac{1}{2})=4(\frac{1}{2})^2-4(\frac{1}{2})+2 \\ =4(\frac{1}{4})-2+2 \\ =1-2+2 \\ =1 \end{gathered}$

Therefore:

$(g\circ f)(\frac{1}{2})=1$