Problem 58 - Algebra for College Students

Question

(a) find \(f^{-1}\) and (b) graph \(f\) and \(f^{-1}\) on the same set of axes. $$ f(x)=\sqrt{x-3} \text { for } x \geq 3 $$

Step-by-Step Solution

Answer

The inverse is \( f^{-1}(x) = x^2 + 3 \).

Step 1: Understand the Formula given

The function is given as \( f(x) = \sqrt{x-3} \) and it is defined for \( x \geq 3 \). This means that the domain of \( f(x) \) is \( x \geq 3 \) and the range is \( y \geq 0 \).

Step 2: Find the Inverse Function

To find the inverse, we swap \( x \) and \( y \), and solve for \( y \). Start with the equation \( y = \sqrt{x-3} \). Replace \( y \) with \( x \) and \( x \) with \( y \) to get: \( x = \sqrt{y-3} \). Squaring both sides, we get \( x^2 = y - 3 \). Solve for \( y \) by adding 3 to both sides: \( y = x^2 + 3 \). The inverse function is \( f^{-1}(x) = x^2 + 3 \), with the domain \( x \geq 0 \).

Step 3: Graph the Original Function \( f(x) \)

The graph of \( f(x) = \sqrt{x-3} \) starts at (3, 0) and increases slowly to the right, forming a curve that looks like a half parabola opening sideways. The lowest point on the graph is the point (3, 0), and it increases as \( x \) increases.

Step 4: Graph the Inverse Function \( f^{-1}(x) \)

The graph of \( f^{-1}(x) = x^2 + 3 \) is a parabola that starts at the point (0, 3) and opens upwards. This is a standard parabola shifted 3 units up and is only plotted for \( x \geq 0 \).

Step 5: Graph Both Functions on the Same Axes

When graphing both functions, \( f(x) \) and \( f^{-1}(x) \), on the same set of axes, they should intersect on the line \( y = x \). The function \( f(x) \) will stretch out to the right starting at (3, 0), and \( f^{-1}(x) \) will be a parabola starting at (0, 3).

Key Concepts

Graphing Functions

When graphing functions, it's important to visually represent how they behave over their domain.The function \(f(x) = \sqrt{x-3}\) has a unique curve that starts at \((3, 0)\) and only increases from there.This is because it is only defined for \(x \geq 3\).

• The graph resembles half of a parabola lying on its side.
• It gradually extends to the right and upwards as \(x\) increases.

Additionally, graphing the inverse function, \(f^{-1}(x) = x^2 + 3\), gives insight into how these two functions relate.The inverse is a standard upward-opening parabola, shifted up by 3 units, again confirming specific values and redefining the coordinate relationships.By plotting both \(f(x)\) and \(f^{-1}(x)\) on the same axes, a clear relationship between the original and its inverse is shown, intersecting along \(y=x\).

Domain and Range

Understanding the domain and range of a function helps define where the function is applicable and what values it can take.In our given function \(f(x) = \sqrt{x-3}\), the domain is \(x \geq 3\).This means the function only exists for values of \(x\) greater than or equal to 3.
The range of this function is \(y \geq 0\), implying that all output values \(y\) are non-negative.

• Domain: Possible input values.
• Range: Possible output values.

For the inverse function \(f^{-1}(x) = x^2 + 3\), the domain and range swap from the original function.The inverse function has a domain \(x \geq 0\) (borrowed from the range of the original function) and a range of \(y \geq 3\).This switch illustrates a fundamental property of inverse functions, where the domain of the original function becomes the range of the inverse and vice versa.

Quadratic Functions

Quadratic functions often appear as parabolas when graphed.A simple form of a quadratic function is \(y = x^2\), which reaches its vertex at the origin and opens upwards.
For our problem, the inverse function \(f^{-1}(x) = x^2 + 3\) is a translated version of this basic quadratic.

• Standard quadratic: \(y = x^2\)
• Translated quadratic: \(f^{-1}(x) = x^2 + 3\)

This translation moves the standard parabola up by 3 units on the y-axis, resulting in a vertex at \((0, 3)\) instead of \((0, 0)\).Such a shift directly impacts the range, lifting the entire parabola above the x-axis.Moreover, we only consider values \(x \geq 0\) for the inverse, ensuring it represents a function that is indeed the reflection of the original when represented together with \(f(x)\).