Problem 46 - 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. $$g(x)=x^{2}-6, x \geq 0$$

Step-by-Step Solution

Answer

The inverse of the function \(g(x) = x^{2} - 6\) when \(x \geq 0\) is \(g^{-1}(x) = \sqrt{x + 6}\). The graphs of the function and its inverse are mirror images about the line \(y = x\). The function \(g(x)\) is a parabola opening upwards starting at \((0,-6)\), and the inverse function \(g^{-1}(x)\) is the right half of a parabola opening to the right starting at \((-6, 0)\).

Step 1 : Rewrite the Function

Rewrite the function \(g(x) = x^{2} - 6\) as \(y = x^{2} - 6\)

Step 2 : Swap x and y

Swap \(x\) and \(y\) to get \(x = y^{2} - 6\)

Step 3 : Solve for y

To solve for \(y\), get all the terms with \(y\) on one side of the equation and the constant terms on the other side. This gives \(y^2 = x + 6\). To get the inverse \(g^{-1}(x)\), take the square root of both sides. The original function was restricted to \(x \geq 0\), so we don't need to consider negative square roots. Thus, the inverse is \(g^{-1}(x) = \sqrt{x + 6}\).

Step 4 : Graph the Functions

Graph the function \(g(x) = x^2 - 6\) and its inverse \(g^{-1}(x) = \sqrt{x + 6}\) on the same set of axes. The function \(g(x) = x^2 - 6\) is a parabola opening upwards starting at \( (0,-6)\) and the inverse function \(g^{-1}(x) = \sqrt{x + 6}\) is the right half of a parabola opening to the right starting at \((-6, 0)\). Because they are inverses, they should be mirror images of each other along the line \(y = x\).

Key Concepts

Graphing Functions

When graphing functions, we often focus on how the equation of the function translates onto a coordinate grid. Graphs provide a visual representation of the relationships between variables in a function. For example, the graph of a quadratic function like \( g(x) = x^2 - 6 \) is a parabola.
This means it has a distinct U-shape that either opens upwards or downwards. In this case, since the coefficient of \(x^2\) is positive, it opens upwards.
When graphing the function and its inverse, it's crucial to remember that they will reflect over the line \( y = x \). In this context, the graph of the inverse function \( g^{-1}(x) = \sqrt{x + 6} \) will appear as a horizontal opening parabola along the positive x-direction.
Here are a few bullet points to remember while graphing functions:

• Identify the shape of the function first. Is it linear, quadratic, or another form?
• Note the starting point and orientation of the graph's opening.
• Graph inverses by reflecting over the line \( y = x \).

Quadratic Functions

Quadratic functions are a core part of algebra, showing up in various forms and scenarios. They are usually written as \( ax^2 + bx + c \), where \(a\), \(b\), and \(c\) are constants. In the exercise, the function \( g(x) = x^2 - 6 \) is a specific type of quadratic function known as a standard form because it lacks the linear component \(bx\).
Quadratics have several key features: the vertex, axis of symmetry, and direction of the parabola. The vertex of \( g(x) = x^2 - 6 \) is at the point \((0, -6)\), and the axis of symmetry is the y-axis.
Quadratics can be modified using transformations, including shifting, reflecting, and stretching. This particular quadratic only shifts vertically by 6 units downward.

• Quadratic graphs are symmetrical around their axis of symmetry.
• The vertex is the turning point of the parabola.
• Direction is determined by the sign of the leading coefficient \( a \).

Function Transformations

Function transformations involve changing the position, size, or orientation of a graph. These transformations can affect the function's domain and range, as seen with inverse functions. In one transformation, the step from \( y = x^2 - 6 \) to \( g^{-1}(x) = \sqrt{x + 6} \) involves finding the inverse, which flips the role of inputs and outputs.
Common transformations include translations (shifts), reflections, stretches, and compressions. For function \( g(x) = x^2 - 6 \):

• Vertical Shift: The graph moves 6 units down.
• Inverse Transformation: Swaps x and y and involves solving for y to reflect over \(y = x\).
• Domain Restrictions: Ensures the original function only includes non-negative x-values for the inverse to be valid.

Understanding these transformations allows you to manipulate and recognize functions graphically, making it easier to predict changes and identify the inverse function’s behavior. Transformations play a significant role in simplifying complex equations and verifying inverse relationships visually.