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Chapter 10: Problem 9

Let \(j(x)=x^{2}\) and \(k(x)=x^{3} .\) Does \(k(j(x))=j(k(x))\) for all values of \(x ?\) Explain.

Short Answer

Expert verified
Yes, \(k(j(x)) = j(k(x))\) for all values of \(x\).

Step by step solution

01

Define the Functions

Identify the given functions: ewlineFor the function \(j(x)\), it is defined as \(j(x) = x^{2}\).ewlineFor the function \(k(x)\), it is defined as \(k(x) = x^{3}\).
02

Find \(k(j(x))\)

Substitute \(j(x)\) into \(k(x)\): ewline\(k(j(x)) = k(x^{2})\). ewlineSince \(k(x) = x^{3}\), we substitute \(x^{2}\) into \(k(x)\):ewlineThis gives us: \[ k(j(x)) = (x^{2})^{3} \]ewlineSimplify the expression: \[ k(j(x)) = x^{6} \]
03

Find \(j(k(x))\)

Substitute \(k(x)\) into \(j(x)\): ewline\(j(k(x)) = j(x^{3})\). ewlineSince \(j(x) = x^{2}\), we substitute \(x^{3}\) into \(j(x)\):ewlineThis gives us: \[ j(k(x)) = (x^{3})^{2} \]ewlineSimplify the expression: \[ j(k(x)) = x^{6} \]
04

Compare \(k(j(x))\) and \(j(k(x))\)

Compare the results from the previous steps: ewline\(k(j(x)) = x^{6}\) and \(j(k(x)) = x^{6}\).ewlineSince \(k(j(x)) = j(k(x))\) for all values of \(x\), the given statement is true.

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Key Concepts

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

Composing functions
Composing functions means performing an operation where the output of one function becomes the input of another function. This results in a new function. In our exercise, we are given two functions, \( j(x) = x^{2} \) and \( k(x) = x^{3} \). We need to determine whether \( k(j(x)) = j(k(x)) \) for all values of \( x \). This involves substituting one function into the other:
  • For \( k(j(x)) \), we first find \( j(x) \), then substitute it into \( k(x) \).
  • For \( j(k(x)) \), we reverse the process and substitute \( k(x) \) into \( j(x) \).
By clearly understanding the steps of substitution and simplification, we can determine if the composite functions are equal.
Function operations
Function operations involve composing, adding, subtracting, multiplying, and dividing functions. In this exercise, our focus is on composing two functions, \( j(x) \) and \( k(x) \). We need to perform the following steps:
  • Define the given functions.
  • Substitute one function into the other to form a composite function.
  • Simplify the expressions.
  • Compare the resulting functions.
By following these steps, we can determine whether the composition of functions remains the same regardless of the order in which they are composed. Understanding function operations is crucial in various areas of mathematics.
Polynomial functions
A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The functions \( j(x) = x^{2} \) and \( k(x) = x^{3} \) are polynomial functions. When composing polynomial functions:
  • We follow the algebraic rules for operations involving exponents.
  • For \( k(j(x)) \), we substitute \( j(x) = x^{2} \) into \( k(x) = x^{3} \), resulting in \( k(j(x)) = (x^{2})^{3} = x^{6} \).
  • Similarly, for \( j(k(x)) \), we substitute \( k(x) = x^{3} \) into \( j(x) = x^{2} \), leading to \( j(k(x)) = (x^{3})^{2} = x^{6} \).
Therefore, both composed functions simplify to \( x^{6} \), proving that the functions are equal for all values of \( x \). This illustrates how polynomial functions behave under composition.
Mathematical proof
A mathematical proof is a logical argument demonstrating the truth of a mathematical statement. In our exercise, we prove that \( k(j(x)) = j(k(x)) \) for all values of \( x \) using straightforward steps:
  • First, define the functions \( j(x) = x^{2} \) and \( k(x) = x^{3} \).
  • Next, substitute \( j(x) \) into \( k(x) \) to find \( k(j(x)) = (x^{2})^{3} = x^{6} \).
  • Then, substitute \( k(x) \) into \( j(x) \) to find \( j(k(x)) = (x^{3})^{2} = x^{6} \).
  • Finally, compare the results. Since both are equal, we conclude that the statement is true.
This method of substitution, simplification, and comparison forms the basis of many mathematical proofs, providing clear and rigorous validations of mathematical statements.

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Most popular questions from this chapter

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

Automobile mufflers are designed to reduce exhaust noise in part by applying wave interference. The resonating chamber of a muffler contains a specific volume of air and has a specific length that is calculated to produce a wave that cancels out a certain frequency of sound. Suppose the engine noise can be modelled by \(E(t)=10 \sin 480 \pi t\) and the resonating chamber produces a wave modelled by \(R(t)=8 \sin 480 \pi(t-0.002),\) where \(t\) is the time, in seconds. a) Graph \(E(t)\) and \(R(t)\) using technology for a time period of 0.02 s. b) Describe the general relationship between the locations of the maximum and minimum values of the two functions. Will this result in destructive interference or constructive interference? c) Graph \(E(t)+R(t)\).

In general, two functions \(f(x)\) and \(g(x)\) are inverses of each other if and only if \(f(g(x))=x\) and \(g(f(x))=x .\) Verify that the pairs of functions are inverses of each other. a) \(f(x)=5 x+10\) and \(g(x)=\frac{1}{5} x-2\) b) \(f(x)=\frac{x-1}{2}\) and \(g(x)=2 x+1\) c) \(f(x)=\sqrt[3]{x+1}\) and \(g(x)=x^{3}-1\) d) \(f(x)=5^{x}\) and \(g(x)=\log _{5} x\)

Use the functions \(f(x)=3 x, g(x)=x-7\), and \(h(x)=x^{2}\) to determine each of the following. a) \((f \circ g \circ h)(x)\). b) \(g(f(h(x)))\) c) \(f(h(g(x)))\) d) \((h \circ g \circ f)(x)\)

For each pair of functions, \(f(x)\) and \(g(x)\), determine \(f(g(x))\) and \(g(f(x))\). a) \(f(x)=x^{2}+x\) and \(g(x)=x^{2}+x\) b) \(f(x)=\sqrt{x^{2}+2}\) and \(g(x)=x^{2}\) c) \(f(x)=|x|\) and \(g(x)=x^{2}\)
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