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Chapter 1: Problem 203

Use composition to determine which pairs of functions are inverses. \(f(x)=\frac{1}{x-1}, x \neq 1, g(x)=\frac{1}{x}+1, x \neq 0\)

Short Answer

Expert verified
Functions \( f(x) = \frac{1}{x-1} \) and \( g(x) = \frac{1}{x} + 1 \) are inverses.

Step by step solution

01

Identify Functions to Check for Inverse

Given functions are \( f(x) = \frac{1}{x-1} \) and \( g(x) = \frac{1}{x} + 1 \). We need to check if these functions are inverses of each other by composing them.
02

Compose f and g: Find f(g(x))

First, substitute \( g(x) = \frac{1}{x} + 1 \) into \( f \). This means we replace every \(x\) in \(f(x)\) with \(g(x)\):\[f(g(x)) = f\left(\frac{1}{x} + 1\right) = \frac{1}{\left(\frac{1}{x} + 1\right) - 1} = \frac{1}{\frac{1}{x}} = x.\]Since this results in \(x\), the composition is correct up to this point.
03

Compose g and f: Find g(f(x))

Next, substitute \( f(x) = \frac{1}{x-1} \) into \( g \). This means we replace every \(x\) in \(g(x)\) with \(f(x)\):\[g(f(x)) = g\left(\frac{1}{x-1}\right) = \frac{1}{\left(\frac{1}{x-1}\right)} + 1 = (x-1) + 1 = x.\]The result is \(x\), so the composition is correct for this part as well.
04

Conclusion: Determine If Functions Are Inverses

Since both compositions \( f(g(x)) = x \) and \( g(f(x)) = x \), the functions are indeed inverses of each other by definition.

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Key Concepts

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

Composition of Functions
Composition of functions is a foundational concept in mathematics. It involves combining two functions where one function鈥檚 output becomes the input for another function. This is denoted as \(f(g(x))\) or \(g(f(x))\). When determining if two functions are inverses, we use composition to check if applying one function after the other returns the original input.
To start composing functions, you always replace every instance of the variable in the first function with the entire output expression of the second function. This means substituting the inner function completely into the outer function wherever its variable is mentioned.
  • For example, with \(f(x)\) and \(g(x)\) given, check \(f(g(x))\).
  • Similarly, check \(g(f(x))\) to verify both compositions return the original input \(x\).
Function Substitution
Function substitution is integral to evaluating composition of functions. It involves plugging one entire function into another. This means every occurrence of the variable in your function will be replaced by the other function鈥檚 expression.
When substituting \(g(x) = \frac{1}{x} + 1\) into \(f(x) = \frac{1}{x-1}\), you replace \(-1\) but only after substituting \(g(x)\) in place of \(x\) in \(f\). This allows effective evaluation of \(f(g(x))\).
  • It's crucial to approach function substitution methodically. Start by writing down the expression needing substitution, then carefully replace all variables.
  • Always simplify the expression after substitution to ensure clarity and correctness.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying expressions to verify mathematical relations, like checking if composed functions are inverses. This requires knowing basic algebra techniques including simplifying fractions, distributing, and combining like terms.
In our exercise, after substituting the functions, we simplify to confirm the output expression simplifies to \(x\). This is essential to establishing that the functions are inverses because both outputs returned the original variable.
  • For instance, once you substitute, balance the equation by clearing out complex fractions or unnecessary terms.
  • Remember that the goal is to confirm that both \(f(g(x))\) and \(g(f(x))\) resolve neatly to \(x\).
  • Careful and methodical manipulation helps prevent mistakes and assures the solution is correct.

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Most popular questions from this chapter

The concentration of hydrogen ions in a substance is denoted by \(\left[\mathrm{H}^{+}\right],\) measured in moles per liter. The pH of a substance is defined by the logarithmic function \(\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right] .\) This function is used to measure the acidity of a substance. The pH of water is \(7 .\) A substance with a pH less than 7 is an acid, whereas one that has a pH of more than 7 is a base. a. Find the \(\mathrm{pH}\) of the following substances. Round answers to one digit. b. Determine whether the substance is an acid or a base. i. Eggs: \(\left[\mathrm{H}^{+}\right]=1.6 \times 10^{-8} \mathrm{mol} / \mathrm{L}\) ii. Beer: \(\left[\mathrm{H}^{+}\right]=3.16 \times 10^{-3} \mathrm{molL}\) iii. Tomato Juice: \(\left[\mathrm{H}^{+}\right]=7.94 \times 10^{-5} \mathrm{molL}\)

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For the following exercises, write the equation in equivalent logarithmic form. $$ 9^{y}=150 $$

ITI A car was purchased for \(\$ 26,000\) . The value of the car depreciates by \(\$ 1500\) per year. a. Find a linear function that models the value \(V\) of the car after \(t\) years. b. Find and interpret \(V(4)\)

For the following exercises, use the change-of-base formula and either base 10 or base \(e\) to evaluate the given expressions. Answer in exact form and in approximate form, rounding to four decimal places. $$ \log _{0.5} 211 $$
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