Problem 92
Question
$$ \text { If } f(x)=x^{2}-\frac{1}{x^{2}}, \text { prove that } f(x)=-f\left(\frac{1}{x}\right) \text { . } $$
Step-by-Step Solution
Verified Answer
We are given the function \(f(x) = x^{2} - \frac{1}{x^{2}}\). We substitute \(x\) with \(\frac{1}{x}\) and simplify to find \(f\left(\frac{1}{x}\right) = \frac{1}{x^{2}} - x^{2}\). Comparing the functions, we find that \(f(x) = -f\left(\frac{1}{x}\right)\), thus proving the property.
1Step 1: Write down the given function
We are provided with the function \(f(x) = x^{2} - \frac{1}{x^{2}}\).
2Step 2: Substitute \(x\) with \(\frac{1}{x}\)
We need to find the value of the \(f\left(\frac{1}{x}\right)\). Let's substitute \(x\) with \(\frac{1}{x}\) in the given function:
\(f\left(\frac{1}{x}\right) = \left(\frac{1}{x}\right)^{2} - \frac{1}{\left(\frac{1}{x}\right)^{2}}\).
3Step 3: Simplify \(f\left(\frac{1}{x}\right)\)
Now we need to simplify the expression for \(f\left(\frac{1}{x}\right)\):
\(f\left(\frac{1}{x}\right) = \frac{1}{x^{2}} - \frac{1}{\frac{1}{x^{2}}}\)
To further simplify the second term, we can multiply the numerator and denominator of the fraction \(\frac{1}{\frac{1}{x^{2}}}\) by \(x^{2}\), which will result in:
\(f\left(\frac{1}{x}\right) = \frac{1}{x^{2}} - x^{2}\).
4Step 4: Compare \(f(x)\) and \(f\left(\frac{1}{x}\right)\)
Now let's compare our original function \(f(x) = x^{2} - \frac{1}{x^{2}}\) with \(f\left(\frac{1}{x}\right) = \frac{1}{x^{2}} - x^{2}\). We will notice that these two functions are negatives of each other: \(f(x) = -f(\frac{1}{x})\).
Therefore, we have proved that \(f(x) = -f\left(\frac{1}{x}\right)\).
Key Concepts
Function SymmetryInverse Function PropertiesSimplifying Mathematical Expressions
Function Symmetry
Understanding the symmetry of mathematical functions is an essential concept in algebra and calculus. A function is said to exhibit symmetry when its graph shows a balanced and mirrored arrangement either across the y-axis, known as even symmetry, or around the origin, known as odd symmetry.
The exercise provided presents a function that displays a particular type of symmetry, odd symmetry. In detail, the function given is \( f(x) = x^2 - \frac{1}{x^2} \). To investigate its symmetry, we seek to establish if \( f(x) \) is equal to \( -f(\frac{1}{x}) \), which would imply that the function is odd. This is demonstrated in the steps of the solution, where the variable \( x \) is replaced with \( \frac{1}{x} \), and upon simplification, it reveals that indeed \( f(x) = -f(\frac{1}{x}) \), hence confirming the odd symmetry.
Understanding function symmetry is crucial not just for proving properties like the one shown, but also for graphing functions and solving equations involving symmetric functions effectively.
The exercise provided presents a function that displays a particular type of symmetry, odd symmetry. In detail, the function given is \( f(x) = x^2 - \frac{1}{x^2} \). To investigate its symmetry, we seek to establish if \( f(x) \) is equal to \( -f(\frac{1}{x}) \), which would imply that the function is odd. This is demonstrated in the steps of the solution, where the variable \( x \) is replaced with \( \frac{1}{x} \), and upon simplification, it reveals that indeed \( f(x) = -f(\frac{1}{x}) \), hence confirming the odd symmetry.
Understanding function symmetry is crucial not just for proving properties like the one shown, but also for graphing functions and solving equations involving symmetric functions effectively.
Inverse Function Properties
When we speak about inverse function properties, we're looking at a relationship that exists between two functions. If we have a function \( f \), its inverse \( f^{-1} \) 'reverses' the effect of \( f \). In terms of equations, for the function \( f(x) \), the inverse \( f^{-1}(y) \) would give us the original value \( x \) back when applied to the output of \( f \), symbolically \( f^{-1}(f(x)) = x \) and \( f(f^{-1}(y)) = y \).
In the example provided, although we are not directly dealing with the formal inverse of the function, the transformation involving the substitution \( x \) with \( \frac{1}{x} \) highlights a related concept. The transformation reflects each point of the graph about the line \( y = x \), which is a critical action when dealing with inverse functions. In practice, verifying the properties of inverse functions often involves a similar process of substitution and simplification to ensure that the functions, in their respective domains, do 'undo' each other.
In the example provided, although we are not directly dealing with the formal inverse of the function, the transformation involving the substitution \( x \) with \( \frac{1}{x} \) highlights a related concept. The transformation reflects each point of the graph about the line \( y = x \), which is a critical action when dealing with inverse functions. In practice, verifying the properties of inverse functions often involves a similar process of substitution and simplification to ensure that the functions, in their respective domains, do 'undo' each other.
Simplifying Mathematical Expressions
Simplifying mathematical expressions involves rewriting them in a more concise and understandable form while retaining their original value. It's a fundamental skill in algebra that helps with solving equations, understanding function behavior, and performing calculus operations more efficiently.
In the exercise, the simplification process is evident in steps 3 and 4. When \( f\left(\frac{1}{x}\right) \) is simplified, the process involves operations such as squaring the reciprocal of \( x \), and then recognizing that \( \frac{1}{\frac{1}{x^2}} = x^2 \) by multiplying the numerator and denominator by \( x^2 \). This simplification is crucial to compare the original function \( f(x) \) and the transformed function \( f\left(\frac{1}{x}\right) \) effectively, which leads to the conclusion that the function has odd symmetry. It's important for students to practice simplifying expressions to facilitate easier handling of more complex mathematical problems.
In the exercise, the simplification process is evident in steps 3 and 4. When \( f\left(\frac{1}{x}\right) \) is simplified, the process involves operations such as squaring the reciprocal of \( x \), and then recognizing that \( \frac{1}{\frac{1}{x^2}} = x^2 \) by multiplying the numerator and denominator by \( x^2 \). This simplification is crucial to compare the original function \( f(x) \) and the transformed function \( f\left(\frac{1}{x}\right) \) effectively, which leads to the conclusion that the function has odd symmetry. It's important for students to practice simplifying expressions to facilitate easier handling of more complex mathematical problems.
Other exercises in this chapter
Problem 90
$$ \text { If } f(x)=\sin x \text { and } g(x)=x^{2}, \text { then find } f \circ g(x) \text { and } \operatorname{gof}(x) \text { . } $$
View solution Problem 91
$$ \text { If } f(x)=x \cos x \text { and } g(x)=\frac{x}{1+x^{2}} \text { , then find } f \circ g(x) \text { and } g o f(x) $$
View solution Problem 93
$$ \text { If } f(x)=x+\frac{1}{x}, \text { prove that }(f(x))^{3}=f\left(x^{3}\right)+3 f\left(\frac{1}{x}\right) \text { . } $$
View solution Problem 94
$$ \text { Given the function } f(x)=\frac{a^{x}+a^{-x}}{2},(a>0), \text { show that } f(x+y)+f(x-y)=2 f(x) f(y) \text { . } $$
View solution