Problem 2
Question
Christopher said that \(\mathrm{f}(x)=|x-2|\) and \(\mathrm{g}(x)=|x+2|\) are inverse functions after he showed that \(\mathrm{f}(\mathrm{g}(2))=2, \mathrm{f}(\mathrm{g}(5))=5,\) and \(\mathrm{f}(\mathrm{g}(7))=7 .\) Do you agree that \(\mathrm{f}\) and \(\mathrm{g}\) are inverse functions? Explain why or why not.
Step-by-Step Solution
Verified Answer
No, \( f(x) = |x - 2| \) and \( g(x) = |x + 2| \) are not inverse functions as they do not satisfy \( f(g(x)) = x \) for all \( x \).
1Step 1: Understand Inverse Functions
Inverse functions are two functions, say \( f \) and \( g \), such that for every value of \( x \) in the domain, \( g(f(x)) = x \) and \( f(g(x)) = x \). This means that applying function \( f \) and then \( g \) (or vice versa) will return the original input.
2Step 2: Analyze Given Functions
Given \( f(x) = |x - 2| \) and \( g(x) = |x + 2| \). Functions \( f \) and \( g \) involve absolute values and shifts. \( f(x) \) shifts \( x \) two units to the right, and \( g(x) \) shifts \( x \) two units to the left, both wrapped in an absolute value.
3Step 3: Test the Inverses
To confirm invertibility, verify that both \( f(g(x)) = x \) and \( g(f(x)) = x \) for all \( x \). First, calculate \( f(g(x)) = f(|x + 2|) = ||x + 2| - 2| \) and see if it simplifies to \( x \).
4Step 4: Simplify and Evaluate \( f(g(x)) \)
Substitute \( g(x) = |x + 2| \) into \( f(x) \), yielding \( f(g(x)) = ||x + 2| - 2| \). This does not always equal \( x \); for example, if \( x = -4 \), \( g(x) = 2 \) and \( f(g(x)) = |2 - 2| = 0 \), not \( -4 \).
5Step 5: Check for Enough Examples
Christopher checked specific values: \( f(g(2))=2, f(g(5))=5, f(g(7))=7 \). However, inverse functions must satisfy the condition \( f(g(x)) = x \) for all \( x \) in the domain, not just certain numbers.
6Step 6: Disprove the Inverse via Counterexample
Provide a counterexample showing \( f(g(x)) eq x \). For \( x = -4 \): \( g(-4) = | -4 + 2 | = 2 \), then \( f(g(-4)) = |2 - 2| = 0 \), not \( -4 \). Thus, \( f \) and \( g \) are not inverses.
Key Concepts
Absolute Value FunctionsFunction CompositionDomain and RangeCounterexampleFunction Transformations
Absolute Value Functions
Absolute value functions are a type of function where the value of the expression inside the absolute value symbol is considered without regard to its sign. This means any negative numbers inside the absolute value become positive.
For example, \(|-3| = 3\) and \(|3| = 3\). Absolute value functions create a V-shaped graph and exhibit certain transformations, such as shifts and reflections.
In the context of the given exercise, both functions \(f(x) = |x - 2|\) and \(g(x) = |x + 2|\) are shifts of the basic absolute value function \(|x|\):
For example, \(|-3| = 3\) and \(|3| = 3\). Absolute value functions create a V-shaped graph and exhibit certain transformations, such as shifts and reflections.
In the context of the given exercise, both functions \(f(x) = |x - 2|\) and \(g(x) = |x + 2|\) are shifts of the basic absolute value function \(|x|\):
- \(f(x)\) shifts the graph two units to the right.
- \(g(x)\) shifts the graph two units to the left.
Function Composition
Function composition involves applying one function to the results of another function. The notation \(f(g(x))\) means you first apply g to \(x\) and then apply \(f\) to the result.
Understanding function composition is essential for determining if two functions are inverses. If \(f\) and \(g\) are inverses, both \(f(g(x)) = x\) and \(g(f(x)) = x\) must hold true for all \(x\) in their domains.
In the exercise, Christopher checked specific values like \(f(g(2)) = 2\), which might give the impression that these functions are inverses.
However, proving invertibility requires examining the composition across the entire domain rather than only a few select values.
Understanding function composition is essential for determining if two functions are inverses. If \(f\) and \(g\) are inverses, both \(f(g(x)) = x\) and \(g(f(x)) = x\) must hold true for all \(x\) in their domains.
In the exercise, Christopher checked specific values like \(f(g(2)) = 2\), which might give the impression that these functions are inverses.
However, proving invertibility requires examining the composition across the entire domain rather than only a few select values.
Domain and Range
The domain of a function is the set of input values \(x\) for which the function is defined, whereas the range is the set of possible output values.
For inverse functions, the domain of \(f\) must match the range of \(g\), and vice versa.
With absolute value functions like \(f(x) = |x - 2|\) and \(g(x) = |x + 2|\), the domain is all real numbers since you can input any real number into an absolute value function.
However, the range of each function affects the invertibility. The output ranges of both functions are all non-negative values since absolute values are never negative. This inherent limitation is essential when considering if these functions can serve as each other's inverses.
For inverse functions, the domain of \(f\) must match the range of \(g\), and vice versa.
With absolute value functions like \(f(x) = |x - 2|\) and \(g(x) = |x + 2|\), the domain is all real numbers since you can input any real number into an absolute value function.
However, the range of each function affects the invertibility. The output ranges of both functions are all non-negative values since absolute values are never negative. This inherent limitation is essential when considering if these functions can serve as each other's inverses.
Counterexample
A counterexample is a specific example that disproves a general statement or claim.
In this exercise, guaranteeing that \(f(g(x)) = x\) and \(g(f(x)) = x\) holds for all values in the functions' domains is necessary to confirm they are inverses.
Christopher's selected examples like \(f(g(2)) = 2\) do not suffice, as just a few instances do not establish the rule.
The counterexample using \(x = -4\) demonstrates \(f(g(-4)) = 0\), not \(-4\). This clearly shows that \(f\) and \(g\) are not inverses, as this one example invalidates the claim for all possible cases.
In this exercise, guaranteeing that \(f(g(x)) = x\) and \(g(f(x)) = x\) holds for all values in the functions' domains is necessary to confirm they are inverses.
Christopher's selected examples like \(f(g(2)) = 2\) do not suffice, as just a few instances do not establish the rule.
The counterexample using \(x = -4\) demonstrates \(f(g(-4)) = 0\), not \(-4\). This clearly shows that \(f\) and \(g\) are not inverses, as this one example invalidates the claim for all possible cases.
Function Transformations
Function transformations involve shifts, stretches, compressions, and reflections of graphs. These transformations can significantly change the appearance and properties of functions.
In the problem at hand, transformations are applied to the basic absolute value function \(|x|\):
These transformations are crucial when determining if functions are inverses as they can influence the composition outcome, especially in functions with similar structures like these.
In the problem at hand, transformations are applied to the basic absolute value function \(|x|\):
- \(f(x) = |x - 2|\) results in a horizontal shift to the right by 2 units.
- \(g(x) = |x + 2|\) shifts the graph to the left by 2 units.
These transformations are crucial when determining if functions are inverses as they can influence the composition outcome, especially in functions with similar structures like these.
Other exercises in this chapter
Problem 2
Explain the difference between direct variation and inverse variation.
View solution Problem 2
Explain why \((x-h)^{2}+(y-k)^{2}=-4\) is not the equation of a circle.
View solution Problem 2
Explain the difference between fg \((x)\) and \(f(g(x))\)
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The graph of \(y=x^{2}-4 x+4\) is tangent to the \(x\) -axis at \(x=2\) and does not intersect the \(x\) -axis at any other point. How many roots does this func
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