Problem 40 - College Algebra with Corequisite Support

Question

For the following exercises, find the inverse of the function and graph both the function and its inverse. $$ f(x)=\frac{1}{x^{2}}, \quad x \geq 0 $$

Step-by-Step Solution

Answer

The inverse of the function is \(f^{-1}(x) = \frac{1}{\sqrt{x}}\) for \(x > 0\).

Step 1: Swap and Set Up the Equation

To find the inverse of the function, start by swapping the roles of \(x\) and \(y\). The function is given as \(f(x) = \frac{1}{x^2}\). We can rewrite it as \(y = \frac{1}{x^2}\) and then swap \(x\) and \(y\) to get \(x = \frac{1}{y^2}\).

Step 2: Solve for y

Now, solve the equation from Step 1 for \(y\). Start by taking the reciprocal of both sides:\(y^2 = \frac{1}{x}\).Next, take the square root of both sides to solve for \(y\):\(y = \pm \sqrt{\frac{1}{x}}\).

Step 3: Choose the Correct Solution

Since \(x \geq 0\) in the original function, and \(f(x) = \frac{1}{x^2}\) is positive, we eliminate the negative square root. Thus, the inverse function is \(f^{-1}(x) = \frac{1}{\sqrt{x}}\), where \(x > 0\).

Step 4: Verify the Inverse Function

Verify that \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\) to ensure that the found inverse is correct. For \(f(f^{-1}(x))\): \(f\left(\frac{1}{\sqrt{x}}\right) = \frac{1}{(\frac{1}{\sqrt{x}})^2} = x\).For \(f^{-1}(f(x))\):\(f^{-1}\left(\frac{1}{x^2}\right) = \frac{1}{\sqrt{\frac{1}{x^2}}} = x\).

Step 5: Graph the Function and Its Inverse

Plot the graph of \(f(x) = \frac{1}{x^2}\), which is a downward-opening curve in the quadrant where \(x \geq 0\). The graph will approach the x-axis as x increases.Graph the inverse function \(f^{-1}(x) = \frac{1}{\sqrt{x}}\), which is an upward-opening curve in the first quadrant. Both the function and its inverse are symmetrical about the line \(y=x\).

Key Concepts

Graphing Functions

Graphing functions gives us a visual representation of mathematical equations. In this exercise, we need to graph both the function and its inverse. Our function is given as \( f(x) = \frac{1}{x^2} \) for \( x \geq 0 \). This means, we only consider positive \( x \)-values, which restricts the graph to the first and second quadrants.

• The curve will descend as \( x \) gets larger.
• The function will never touch the x-axis or the y-axis as those are horizontal and vertical asymptotes.

The inverse, \( f^{-1}(x) = \frac{1}{\sqrt{x}} \), is graphed under the assumption \( x > 0 \). This results in an upward curve:

• When \( x \) is small, \( f^{-1}(x) \) starts steeply, approaching the y-axis closely.
• As \( x \) increases, the curve smooths out, having the x-axis as an asymptote.

By graphing both, we can see that functions can take various shapes and behaviors, depending on their mathematical formulation.

Function Verification

When we find an inverse function, it is crucial to verify it. This ensures that both functions are indeed inverses of each other. In our case, for the original function \( f(x) = \frac{1}{x^2} \) and its inverse \( f^{-1}(x) = \frac{1}{\sqrt{x}} \), we must check two conditions:

• \( f(f^{-1}(x)) = x \)
• \( f^{-1}(f(x)) = x \)

Let's verify:1. Substitute \( f^{-1}(x) \) into \( f(x) \), which gives \( f\left(\frac{1}{\sqrt{x}}\right) = \frac{1}{(\frac{1}{\sqrt{x}})^2} = x \).2. Substitute \( f(x) \) into \( f^{-1}(x) \), which gives \( f^{-1}\left(\frac{1}{x^2}\right) = \frac{1}{\sqrt{\frac{1}{x^2}}} = x \).Both equations return \( x \), confirming the correctness of our inverse. Verification guarantees that inverses reliably reverse the operations of functions, providing a vital check in mathematical processes.

Quadrants

In the Cartesian plane, quadrants help us determine where a function or its parts are positioned. The function \( f(x) = \frac{1}{x^2} \) is defined only for \( x \geq 0 \), generally placing its graph mostly in the first quadrant.

• The first quadrant is where both x and y values are positive.

When graphing the inverse \( f^{-1}(x) = \frac{1}{\sqrt{x}} \), it's also constrained to the first quadrant since \( x > 0 \) and results are positive.Quadrants in graphing help students visualize complex or mundane equations, by providing geographical clues as to where curve parts might lie, based on positive or negative inputs and outputs.

Symmetry in Functions

Symmetry can significantly simplify graph interpretations. For functions and their inverses, often the symmetry line is \( y = x \). This means:

• The function and its inverse will reflect symmetrically across the line \( y = x \).

For \( f(x) = \frac{1}{x^2} \) and its inverse \( f^{-1}(x) = \frac{1}{\sqrt{x}} \), this symmetry highlights that for every point \( (a, b) \) on the function's graph, there is a corresponding point \( (b, a) \) on the inverse.Understanding this symmetry:

• Helps easily visualize how inverse functions form in relation to each other.
• Simplifies plotting inverse functions by reflecting over \( y = x \).

Ultimately, symmetry fosters deeper insights into the relationship between functions and their inverses, facilitating more intuitive graphing and analysis.