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Chapter 8: Problem 20

In Exercises \(16-22,\) show that the two functions are inverses of each other. $$ f(x)=10^{x / 2} \text { and } g(x)=\log \left(x^{2}\right) $$

Short Answer

Expert verified
Since \(f(g(x)) = x\) and \(g(f(x)) = x\), the functions are inverses.

Step by step solution

01

- Understand Inverse Functions

To show that two functions are inverses of each other, verify that composing one with the other returns the input value. Specifically, check if \(f(g(x)) = x\) and \(g(f(x)) = x\).
02

- Find \(f(g(x))\)

First, substitute \(g(x)\) into \(f(x)\): \[ f(g(x)) = f(\log(x^2)) \]. Now, replace \(x\) in \(f(x)\) with \( \log(x^2) \): \[ f(\log(x^2)) = 10^{(\log(x^2)) / 2} \]. Using the property of logarithms \( \log(a^b) = b \log(a) \), it simplifies to \[ 10^{(2 \log(x)) / 2} = 10^{\log(x)} \]. Knowing that \(10^{\log(x)} = x\), we conclude \[ f(g(x)) = x \].
03

- Find \(g(f(x))\)

Next, substitute \(f(x)\) into \(g(x)\): \[ g(f(x)) = g(10^{x/2}) \]. Now, replace \(x\) in \(g(x)\) with \(10^{x/2}\): \[ g(10^{x/2}) = \log((10^{x/2})^2) \]. Simplify the expression inside the logarithm: \[ \log(10^x) \]. Using the property \(\log(10^a) = a\), it simplifies to \[ x \]. So, \[ g(f(x)) = x \].
04

- Conclusion

Since both \(f(g(x)) = x\) and \(g(f(x)) = x\), \(f(x) = 10^{x/2}\) and \(g(x) = \log(x^2)\) are inverses of each other.

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Key Concepts

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

Function Composition
Before diving into whether two functions are inverses, it is important to understand function composition. Function composition means plugging one function into another. If we have functions \(f\) and \(g\), the composition \(f(g(x))\) means we first apply \(g(x)\) and then apply \(f\) to that result. To show that two functions are inverses of each other, we need to check that \(f(g(x)) = x\) and \(g(f(x)) = x\). This means doing the operation of \(g\) followed by the operation of \(f\) should give back the original input \(x\), and vice versa.
Logarithmic Properties
Logarithms are the 'opposite' of exponents, making them crucial in solving equations involving exponentials. One key property to remember is \(\text{log}(a^b) = b \text{log}(a)\). This property helps us simplify many complex expressions. Another important property is that a number raised to a logarithm with the same base cancels out, e.g., \(10^{\text{log}(x)} = x\). Using these properties, we can simplify complex compositions involving logarithms. Understanding these fundamental properties will help in simplifying and verifying steps in the exercise.
Algebraic Verification
Algebraic verification is about confirming the work done through algebraic manipulation. In our exercise, we verify that \(f\) and \(g\) are inverses by showing two main results: \(f(g(x)) = x\) and \(g(f(x)) = x\). For example, to find \(f(g(x))\), substitute \(g(x) = \text{log}(x^2)\) into \(f(x)\). After simplifying, we use logarithm properties to simplify \(10^{(\text{log}(x^2))/2}\) to \(x\). Similarly, verify \(g(f(x))\) by substituting \(f(x) = 10^{x/2}\) into \(g(x)\) and simplifying using the properties mentioned. Ensuring both functions simplify back to \(x\) confirms they are indeed inverses.

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Most popular questions from this chapter

(Graphing program required.) At low speeds an automobile engine is not at its peak efficiency; efficiency initially rises with speed and then declines at higher speeds. When efficiency is at its maximum, the consumption rate of gas (measured in gallons per hour) is at a minimum. The gas consumption rate of a particular car can be modeled by the following equation, where \(G\) is the gas consumption rate in gallons per hour and \(M\) is speed in miles per hour: \(G=0.0002 M^{2}-0.013 M+1.07\) a. Construct a graph of gas consumption rate versus speed. Estimate the minimum gas consumption rate from your graph and the speed at which it occurs. b. Using the equation for \(G,\) calculate the speed at which the gas consumption rate is at its minimum. What is the minimum gas consumption rate? c. If you travel for 2 hours at peak efficiency, how much gas will you use and how far will you go? d. If you travel at \(60 \mathrm{mph}\), what is your gas consumption rate? How long does it take to go the same distance that you calculated in part (c)? (Recall that travel distance = speed \(\times\) time traveled.) How much gas is required for the trip? e. Compare the answers for parts (c) and (d), which tell you how much gas is used for the same-length trip at two different speeds. Is gas actually saved for the trip by traveling at the speed that gives the minimum gas consumption rate? f. Using the function \(G,\) generate data for gas consumption rate measured in gallons per mile by completing the following table. Plot gallons per mile (on the vertical axis) vs. miles per hour (on the horizontal axis). At what speed is gallons per mile at a minimum? g. Add a fourth column to the data table. This time compute miles/gal \(=\mathrm{mph} /(\mathrm{gal} / \mathrm{hr}) .\) Plot miles per gallon vs. miles per hour. At what speed is miles per gallon at a maximum? This is the inverse of the preceding question; we are normally used to maximizing miles per gallon instead of minimizing gallons per mile. Does your answer make sense in terms of what you found for parts (b) and (f)?

For the following quadratic functions in vertex form, \(f(x)=a(x-h)^{2}+k,\) determine the values for \(a, h,\) and \(k\) Then compare each to \(f(x)=x^{2},\) and identify which constants represent a stretch/compression factor, or a shift in a particular direction. a. \(p(x)=5(x-4)^{2}-2\) b. \(g(x)=\frac{1}{3}(x+5)^{2}+4\) c. \(h(x)=-0.25\left(x-\frac{1}{2}\right)^{2}+6\) d. \(k(x)=-3(x+4)^{2}-3\)

(Graphing program optional.) Create a quadratic function in the vertex form \(y=a(x-h)^{2}+k,\) given the specified values for \(a\) and the vertex \((h, k) .\) Then rewrite the function in the standard form \(y=a x^{2}+b x+c .\) If available, use technology to check that the graphs of the two forms are the same. a. \(a=1,(h, k)=(2,-4)\) c. \(a=-2,(h, k)=(-3,1)\) b. \(a=-1,(h, k)=(4,3)\) d. \(a=\frac{1}{2},(h, k)=(-4,6)\)

Estimate the maximum number of horizontal intercepts for each of the polynomial functions. Then graph the function using technology to find the actual number. a. \(y=x^{4}-2 x^{2}-5\) b. \(y=4 t^{6}+t^{2}\) c. \(y=x^{3}-3 x^{2}+4\) d. \(y=5+x\)

(Graphing program optional.) a. On separate grids sketch the graphs of \(f(x)=\sqrt{-x+2}\) and \(g(x)=-\sqrt{x+2}\) b. Using interval notation, describe the domains of \(f(x)\) and \(g(x)\) c. Using interval notation, describe the ranges of \(f(x)\) and \(g(x)\) d. What is the simplest function \(h(x)\) from which both \(f(x)\) and \(g(x)\) could be created? e. Describe the transformations of \(h(x)\) to obtain \(f(x)\). Of \(h(x)\) to obtain \(g(x)\) f. Does the graph of \(f(x), g(x),\) or \(h(x)\) have any symmetries (across the \(x\) - or \(y\) -axis, or about the origin)?
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