Question

(a) If \(g\left( x \right) = {\bf{2}}x + {\bf{1}}\) and \(h\left( x \right) = {\bf{4}}{x^{\bf{2}}} + {\bf{4}}x + {\bf{7}}\), find a function f such that \(f \circ g = h\). (Think about what operations you would have to perform on the formula of g to end up with the formula for h.)

(b) If \(f\left( x \right) = {\bf{3}}x + {\bf{5}}\) and \(h\left( x \right) = {\bf{3}}{x^{\bf{2}}} + {\bf{3}}x + {\bf{2}}\), find a function g such that \(f \circ g = h\).

Short answer

(a) \(f\left( x \right) = {x^2} + 6\)

(b) \(g\left( x \right) = {x^2} + x - 1\)

Step-by-step solution

Find the function f

Let the function f as:

\(f\left( x \right) = {x^2} + c\)

Thecomposition\(f \circ g\) can be calculated as:

\(\begin{aligned}f \circ g\left( x \right) &= f\left( {g\left( x \right)} \right)\\ &= f\left( {2x + 1} \right)\\ &= {\left( {2x + 1} \right)^2} + c\\ &= 4{x^2} + 4x + 1 + c\end{aligned}\)

On comparing the composition with \(h\left( x \right)\):

\(\begin{aligned}1 + c &= 7\\c &= 6\end{aligned}\)

So, the function f is given by \(f\left( x \right) = {x^2} + 6\).

Find the function g

The composite function \(f \circ g\) can be calculated as:

\(\begin{aligned}f \circ g\left( x \right) &= f\left( {g\left( x \right)} \right)\\ &= 3g\left( x \right) + 5\\ &= h\left( x \right)\end{aligned}\)

Simpify the function \(h\left( x \right)\).

\(\begin{aligned}h\left( x \right) &= 3{x^2} + 3x + 2\\ &= 3\left( {{x^2} + x - 1} \right) + 5\end{aligned}\)

On comparing the equation with the composition \(f \circ g\) as:

\(g\left( x \right) = {x^2} + x - 1\)

Thus, the function is \(g\left( x \right) = {x^2} + x - 1\).