Question

For each of the following functions,determine the equation of the inverse graph \(f(x)\) and the inverse of \(f(x)\) restrict the domain of \(f(x)\) so that the inverse of \(f(x)\) is a function with the domain of \(f(x)\) restricted, sketch the graphs of \(f(x)\) and \(f^{-1}(x)\). a) \(f(x)=x^{2}+3\) b) \(f(x)=\frac{1}{2} x^{2}\) c) \(f(x)=-2 x^{2}\) d) \(f(x)=(x+1)^{2}\) e) \(f(x)=-(x-3)^{2}\) f) \(f(x)=(x-1)^{2}-2\)

Short answer

Inverses: a) \( \sqrt{x-3} \), b) \( \sqrt{2x} \), c) -\(\sqrt{\frac{-x}{2}} \), d) \( \sqrt{x} - 1 \), e) \( 3-\sqrt{-x} \), f) \( \sqrt{x+3}+1 \)

Step-by-step solution

Identify the type of function

Determine that the given functions are quadratic functions of the form \(f(x) = ax^2 + bx + c\), which are parabolas.

Ensure the function is one-to-one

Restrict the domain of each function so that the function is one-to-one (either decreasing or increasing). This will allow for the existence of an inverse function.

Substitute \(y\) for \(f(x)\)

Rewrite each function in terms of \(y\) instead of \(f(x)\). For example, for part a) we have \(y = x^2 + 3\).

Solve for \(x\) in terms of \(y\)

Express \(x\) in terms of \(y\) in order to find the equation of the inverse. For instance, solve \(y = x^2 + 3\), so \(x = \sqrt{y - 3}\) or \(x = -\sqrt{y - 3}\).

Express the inverse function

Replace \(y\) with \(x\) in the expression found in Step 4 to get the inverse function. E.g., for part a), we get \(f^{-1}(x) = \sqrt{x - 3}\) or \(f^{-1}(x) = -\sqrt{x - 3}\).

Restrict the domain of the original function

Restrict the domain of \(f(x)\) so that the inverse is a function. Typically, this means restricting \(f(x)\) to \(x \geq 0\) or \(x \leq 0\).

Write the final results for each part

Summarize the results for each function and its inverse, including the domain restrictions.

Sketch the graphs

Draw the graphs of each original function and its inverse. The inverse graph will be a reflection of \(f(x)\) over the line \(y=x\).

Key concepts

quadratic functions

Quadratic functions are a specific type of polynomial function where the highest degree of the variable is 2. They take the form \(f(x) = ax^2 + bx + c\), where \(a, b,\) and \(c\) are constants, and \(a eq 0\). These functions create a U-shaped graph known as a parabola. Here we are dealing with different quadratic functions:

• \(f(x) = x^2 + 3\)
• \(f(x) = \frac{1}{2} x^2\)
• \(f(x) = -2 x^2\)
• \(f(x) = (x+1)^2\)
• \(f(x) = -(x-3)^2\)
• \(f(x) = (x-1)^2 - 2\)

Understanding that these functions are quadratic is important for finding their inverse.

one-to-one function

A function is called one-to-one if every value in the range corresponds to exactly one value in the domain. For a function to have an inverse that is also a function, it must be one-to-one. Quadratic functions, by default, are not one-to-one because they are parabolas that open up or down, causing the same y-value to correspond to two x-values. To make a quadratic function one-to-one:

• Restrict the domain to either \(x \geq 0\) or \(x \leq 0\).
• This ensures each y-value is paired with only one x-value, allowing the function's inverse to also be a function without violating the Definition of Functions.

For example, for \(f(x) = x^2 + 3\), restricting the domain to \(x \geq 0\) makes it one-to-one.

function domain restriction

Restricting the domain of a function is essential in finding its inverse when the original function is not one-to-one across its whole domain. For quadratic functions, this involves limiting the x-values. Without restriction, a quadratic function might fail the horizontal line test, which determines if a function is one-to-one.

• Consider \(f(x) = x^2 + 3\). Its unrestricted domain is all real numbers, \((-\infty, \infty)\).
• To make it one-to-one, we can restrict its domain to \(x \geq 0\), making sure each y-value is unique.

This process allows us to find the inverse function, ensuring it remains valid and functional.

inverse graph

The graph of an inverse function is the reflection of the original function's graph over the line \(y = x\). To find the inverse graph for our quadratic functions:

• Firstly, we solve the equation of the function for x in terms of y.
• For \(f(x) = x^2 + 3\), rewriting it as \(y = x^2 + 3\), we solve for \(x\) to get \(x = \sqrt{y - 3}\) and \(x = -\sqrt{y - 3}\).
• We then replace \(y\) with \(x\) to get \(f^{-1}(x) = \sqrt{x - 3}\) and \(f^{-1}(x) = -\sqrt{x - 3}\).

When graphing, the inverse functions will mirror the original functions across the \(y = x\) line. This mirror effect visually demonstrates how every point \((a, b)\) on \(f(x)\) corresponds to \((b, a)\) on \(f^{-1}(x)\).