Question

3. Suppose f(x) = x squared - 2x +3 and g(x) = x squared -3. Compute the composition f(g(x)) and simplify.

Answer 1

The given functions are

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

The composition f(g(x)) refers to substituting the x-variables of f(x) for the function g(x).

$f(g(x))=(x^2-3)^2-2(x^2-3)+3$

Then, we solve the power and product. We solve the squared binomial using the following

$(a-b)^2=a^2-2ab+b^2$$\begin{gathered} f(g(x))=x^4-2\cdot x^2_{}\cdot3+9-2x^2+6+3 \\ f(g(x))=x^4-6x^2-2x^2+18 \\ f(g(x))=x^4-8x^2+18 \end{gathered}$

Therefore, the composition is

$f(g(x))=x^4-8x^2+18$