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Chapter 10: Problem 7

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

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

Step by step solution

01

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 \(f.\)
02

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, \(g(x) = 2x - 5.\)
03

Solution for Part (a)

\(g(x) = 2x - 5\) is the inner function such that \(f(g(x)) = h(x).\)
04

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 \(g(x) = 5x + 1.\)
05

Solution for Part (b)

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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)$
.

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Most popular questions from this chapter

For each pair of functions, \(f(x)\) and \(g(x)\) \(\bullet\) determine \(h(x)=(f \cdot g)(x)\) \(\bullet\) sketch the graphs of \(f(x), g(x),\) and \(h(x)\) on the same set of coordinate axes \(\bullet\) state the domain and range of the combined function \(h(x)\) a) \(f(x)=x^{2}+5 x+6\) and \(g(x)=x+2\) b) \(f(x)=x-3\) and \(g(x)=x^{2}-9\) c) \(f(x)=\frac{1}{x+1}\) and \(g(x)=\frac{1}{x}\)

Consider \(f(x)=x^{2}-9\) and \(g(x)=\frac{1}{x}\) a) State the domain and range of each function. b) Determine \(h(x)=f(x)+g(x)\). c) How do the domain and range of each function compare to the domain and range of \(h(x) ?\).

Given \(f(x)=3 x^{2}+2, g(x)=\sqrt{x+4},\) and \(h(x)=4 x-2,\) determine each combined function and state its domain. a) \(y=(f+g)(x)\) b) \(y=(h-g)(x)\) c) \(y=(g-h)(x)\) d) \(y=(f+h)(x)\)

Given \(f(2)=3, f(3)=4, f(5)=0, g(2)=5\), \(g(3)=2,\) and \(g(4)=-1,\) evaluate the following. a) \(f(g(3))\) b) \(f(g(2))\) c) \(g(f(2))\) d) \(g(f(3))\)

An alternating current-direct current (AC-DC) voltage signal is made up of the following two components, each measured in volts \((\mathrm{V}): V_{\mathrm{AC}}(t)=10 \sin t\) and \(V_{\mathrm{DC}}(t)=15\). a) Sketch the graphs of these two functions on the same set of axes. Work in radians. b) Graph the combined function \(V_{A C}(t)+V_{\mathrm{DC}}(t)\) c) Identify the domain and range of \(V_{\mathrm{AC}}(t)+V_{\mathrm{DC}}(t)\) d) Use the range of the combined function to determine the following values of this voltage signal. i) minimum ii) maximum
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