Problem 55 - Precalculus : Building Concepts and Connections

Question

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$f(x)=\sqrt{x+3}, x \geq-3$$

Step-by-Step Solution

Answer

The inverse function of \(f(x)=\sqrt{x+3}\) is \(f^{-1}(x)=x^2-3\), where \(x \geq 0\)

Step 1: Find the Inverse Function

The given function is \(f(x) = \sqrt{x + 3}\). Remember that \(f(x)\) can be replaced with \(y\). Therefore, the function can be written as \(y = \sqrt{x + 3}\). To find the inverse function, switch the roles of \(x\) and \(y\), thus getting \(x = \sqrt{y + 3}\). Now, solve for \(y\) to express the inverse function in terms of \(x\). Squaring both sides, we get \(x^2 = y + 3\). Solving for \(y\) gives \(y = x^2 - 3\). Therefore, the inverse function is \(f^{-1}(x) = x^2 - 3\), where \(x \geq 0\).

Step 2: Graph the Given Function

Graph the function \(f(x) = \sqrt{x + 3}\). Remember that the radicand, \(x + 3\), has to be non-negative, so \(x\) must be \(-3\) or greater. The domain is \([-3, \infty)\) and the range is \([0, \infty)\). The graph is a basic square root function shifted 3 units to the left.

Step 3: Graph the Inverse Function

Graph the inverse function \(f^{-1}(x) = x^2 - 3\). Note that \(x\) must be \(0\) or greater. The domain is \([0, \infty)\) and the range is \([-3, \infty)\). The graph is a basic quadratic function shifted down 3 units.

Key Concepts

Inverse Function Graphing

Grasping the concept of inverse functions and how to graph them is pivotal for exploring many mathematical relationships. To begin with, an inverse function essentially reverses the operations of the original function. If you consider a function that takes an input and performs some operations to yield an output, the inverse function takes this output as its input and performs the reverse operations to yield the original input.

When graphing inverse functions, an important characteristic to remember is that the graph of an inverse function is a reflection of the graph of the original function over the line y=x. That means every point (a, b) on the graph of the original function will correspond to the point (b, a) on the graph of the inverse function. They are mirror images with respect to the line y=x.

For the provided exercise, you saw the original function was the square root function \(f(x)=\sqrt{x+3}\), and the inverse was obtained as a quadratic function \(f^{-1}(x) = x^2 - 3\). Graphically, these two functions would reflect across the line y=x, showing the domain of one as the range of the other, and vice versa.

Square Root Functions

Square root functions are a type of radical function and they are essential in understanding various mathematical properties, such as the function \(f(x) = \sqrt{x + 3}\) in our exercise. A key characteristic of the square root function is its domain; since you cannot take the square root of a negative number (in the realm of real numbers), the values under the square root sign (the radicand) must be non-negative.

The general form of a square root function is \(f(x) = \sqrt{x - h} + k\), where (h, k) represents a horizontal and vertical shift from the origin, respectively. In our exercise, the function \(f(x)\) is shifted 3 units to the left, as indicated by the +3 inside the radical. As for the graph, it exhibits a characteristic half-parabola shape that starts at the point (-h, k) and extends infinitely to the right.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, often written in the form \(f(x) = ax^2 + bx + c\) where a, b, and c are constants and \(a ≠ 0\). They are ubiquitous in many fields, from physics to finance. The graph of a quadratic function is a parabola that can open upwards or downwards, depending on the sign of the leading coefficient a.

When faced with the inverse of a square root function, as in our exercise, we get a quadratic function \(f^{-1}(x) = x^2 - 3\). This new function represents a parabola opening upwards because the coefficient of \(x^2\) is positive. It is also shifted vertically down by 3 units due to the -3 at the end of the equation. Understanding these transformations is crucial for graphing quadratic functions. The domain and range of quadratic functions are all real numbers, but in the context of being an inverse to the square root function, we restrict the domain of the quadratic to non-negative numbers to maintain the function-inverse relationship.