Question

If \(h(x)=(f \circ g)(x),\) determine \(g(x)\). a) \(h(x)=(2 x-5)^{2}\) and \(f(x)=x^{2}\) b) \(h(x)=(5 x+1)^{2}-(5 x+1)\) and \(f(x)=x^{2}-x\)

Short answer

a) \(g(x) = 2x - 5\) b) \(g(x) = 5x + 1\)

Step-by-step solution

Understanding Composition

Given that \(h(x) = (f \circ g)(x),\) it means \(h(x) = f(g(x)).\) To find \(g(x),\) identify the inner function that fits into the outer function \(f.\)

Determine g(x) for Part (a)

For \(h(x) = (2x - 5)^2\) and \(f(x) = x^2,\) we look for the expression \(g(x)\) such that \(f(g(x)) = h(x).\) Notice that \(f(x) = x^2\) suggests the inner expression \(g(x)\) must satisfy \(g(x)^2 = (2x - 5)^2.\) Thus, \(g(x) = 2x - 5.\)

Solution for Part (a)

\(g(x) = 2x - 5\) is the inner function such that \(f(g(x)) = h(x).\)

Determine g(x) for Part (b)

For \(h(x) = (5x + 1)^2 - (5x + 1)\) and \(f(x) = x^2 - x,\) we need to find \(g(x)\) such that \(f(g(x)) = h(x).\) Notice that \(f(x) = x^2 - x \) means \(g(x)\) must be substituted into both terms. Hence, \(h(x) = f(g(x)) = g(x)^2 - g(x).\) Observing \(h(x),\) recognize that \(g(x) = 5x + 1.\)

Solution for Part (b)

\(g(x) = 5x + 1\) is the inner function such that \(f(g(x)) = h(x).\)

Key concepts

Function Notation

When dealing with functions, it is important to understand function notation. Function notation uses a letter, like \(f\), \(g\), or \(h\), paired with an input value to denote the output of the function for that input. For example, if we have a function \(f(x)=x^2\), the notation \(f(3)\) means we should substitute \(3\) for \(x\), giving us \(f(3)=3^2=9\). This notation helps to clearly define which function we are discussing and the specific value being input into that function.

Inner Function

The term 'inner function' refers to the function that is inside another function in a composition. In the expression \(h(x) = (f \bullet g)(x)\), \(g\) is the inner function. This means \(g(x)\) is substituted first before applying the outer function \(f\). For example, in the problem where \(h(x) = (2x-5)^2\) and \(f(x) = x^2\), the inner function \(g(x) = 2x - 5\) is plugged into \(f\) to produce \(h\):

• \(f(g(x)) = f(2x-5)\)
• \(= (2x-5)^2\)

. Similarly, for \(h(x) = (5x+1)^2 - (5x+1)\) and \(f(x) = x^2 - x\), the inner function is \(g(x) = 5x + 1\).

Outer Function

The 'outer function' in a composition of functions refers to the function that acts on the result of the inner function. Using the notation \(h(x) = (f \bullet g)(x)\), \(f\) is the outer function. This means we first apply the inner function \(g(x)\) and then use this result as the input for the outer function \(f\). For instance, given \(h(x) = (2x - 5)^2\) and understanding that \(g(x) = 2x - 5\), the outer function \(f\) is \(x^2\). Hence,

• \(f(g(x)) = f(2x - 5)\)
• \(= (2x-5)^2\)
• = \(h(x)\)

. The same logic applies to part (b) where \(f(x) = x^2 - x\), and \(g(x) = 5x + 1\). Applying \(g(x)\) to \(f\) yields:

• \(f(g(x)) = (5x+1)^2 - (5x+1)\)
• = h(x)$

.