Problem 45 - 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. $$g(x)=x^{2}-5, x \leq 0$$

Step-by-Step Solution

Answer

The inverse of the function \(g(x) = x^2 - 5, x \leq 0\) is \(g^{-1}(x) = -\sqrt{x + 5}\). The graph of these two functions would show a downward opening parabola for g(x) and a sideways parabola facing towards the negative x-axis for \(g^{-1}(x)\) shifted 5 units to the left.

Step 1: Find the Inverse of the Function

To find the inverse of the function, swap x with y in the equation, so now we'll be solving the equation for x to find \(g^{-1}(x)\). Hence, the equation becomes \(x = y^2 - 5\). Now, to isolate y, add 5 to both sides of the equation to get: \(x + 5 = y^2\). Next, take the square root of both sides (Since y is \( \leq 0\), we take the negative square root):Therefore \(g^{-1}(x) = -\sqrt{x+5}\)

Step 2: Graph the Function and its Inverse

On the same set of axes, we will plot both \(g(x)\) and \(g^{-1}(x)\). When sketching \(g(x) = x^2 - 5\), remember it's a downward opening parabola (because it's \(x^2\) and \(x \leq 0\)) shifted down 5 units on the y-axis.When graphing \(g^{-1}(x) = -\sqrt{x + 5}\), remember that square root functions typically look like half of a sideways parabola with the 'opening' facing towards the positive x (for positive square root) or negative x (for negative square root) axis. In this case, since it's \(-\sqrt{x + 5}\), it will open towards the negative x-axis and shifted 5 units to the left on the x-axis.

Key Concepts

Graphing Functions

Graphing functions is a foundational concept in mathematics that helps us visualize how a function behaves. When you graph a function, you plot each possible input value against its corresponding output value. This creates a visual representation of how inputs (usually represented on the x-axis) relate to outputs (usually on the y-axis). For any function like the one given, each point on the graph represents an input-output pair

• Start by identifying the function you'll be graphing. In the exercise, this was initially given as \( g(x) = x^2 - 5 \).
• Determine key points like vertex, intercepts, or direction of opening which help in sketching the graph.
• Find the function's domain and range. This helps determine the extent of the graph on the axes.
• Plot these points on the coordinate plane.
• Finally, draw a smooth curve through the points to complete the graph.

This graphical method helps in illustrating the nature of the function and is especially crucial when you're dealing with inverse functions.
Both the graph of the function and its inverse can be used to understand how each function 'undoes' the effect of the other.

Parabolas

A parabola is a U-shaped curve that can open in different directions depending on the function representing it. In our exercise, the original function \( g(x) = x^2 - 5 \) is a downward-opening parabola as we only considered values \(x \leq 0\).
This is because squaring any negative number results in a positive value, causing a downward shift on the y-axis.

• The vertex of this parabola is at \((0, -5)\), due to the constant \(-5\).
• The vertex is also the lowest point of the parabola when considering \(x \leq 0\).
• This makes it easy to predict and plot key points for sketching the graph.

Understanding how parabolas work aids in visualizing quadratic functions and inversely allows understanding their transformations when finding an inverse.

Square Root Functions

Square root functions are another important concept in graphing, especially when dealing with inverses. The inverse of a quadratic parabola—which was \( g(x) = x^2 - 5 \) in this case—leads to a square root function. The inverse \( g^{-1}(x) = -\sqrt{x + 5} \) reflects characteristics typical of square root functions but with some key differences due to being the inverse.

• Square root graphs often resemble half of a sideways parabola.
• Since our square root function is \(-\) and \(x + 5\) inside, it shifts and opens leftward. In our case, it shifts 5 units left along the x-axis.
• Square root functions typically only exist for certain values as they are not defined for negative inputs. Here, \(x + 5 \geq 0\) is necessary for \(g^{-1}(x)\).
• The inverse graph completes the picture of how functions inter-relate visually.

Graphing square root functions builds intuition about dealing with their form and understanding how inverses derive from the original quadratic graphs.