Problem 48 - 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. $$g(x)=3 x^{3}-5$$

Step-by-Step Solution

Answer

The inverse function of \(g(x)=3 x^{3}-5\) is \(g^{-1}(x) = \sqrt[3]{((x+5)/3)}\). When plot both the function and its inverse, they should be symmetrical about the line \(y=x\). The result of substituting the inverse function into the original function confirms the correctness.

Step 1: Find Function's Inverse

To find the inverse of the given function \(g(x)\), we need to replace \(g(x)\) with \(y\). So, we have \(y=3 x^{3}-5\). Now, switch the roles of \(y\) and \(x\), gives us \(x = 3y^3-5\). Now, solve this equation for \(y\). Adding 5 to both sides gives \(x+5 = 3y^3\). Then, divide both sides by 3 to isolate \(y^3\) on one side of the equation, hence we get \((x+5)/3 = y^3\). So, \(y = \sqrt[3]{((x+5)/3)}\), which is the inverse function of \(g(x)\), denoted as \(g^{-1}(x)\).

Step 2: Plot function and its Inverse

Now, to plot the function \(g(x)=3 x^{3}-5\) and its inverse \(g^{-1}(x) = \sqrt[3]{((x+5)/3)}\) on the same graph, denote \(g(x)\) by a line plot and \(g^{-1}(x)\) by a dashed line, and be sure to label each one correctly. It's important to note that the function and its inverse should be symmetrical about the line \(y = x\).

Step 3: Check Correctness of Inverse

To confirm that we have identified the correct inverse function, substitute the inverse function into the original function. If we have done everything correctly, we should get \(x\) back as the result. In this case, if we substitute \(g^{-1}(x)\) into \(g(x)\), we get \(3*(\sqrt[3]{((x+5)/3)})^3-5=x\)

Key Concepts

Cubic Functions

A cubic function is a type of polynomial function where the highest degree is three. This means that the variable, usually represented as \(x\), is raised to the third power at its highest. The general form of a cubic function is \(f(x) = ax^3 + bx^2 + cx + d\). Here, \(a\), \(b\), \(c\), and \(d\) are constants, with \(a eq 0\).

• The shape of a cubic function can vary based on the values of these constants. It often has a characteristic "S" shape when plotted.
• Cubic functions may have one or two humps or turns, which are local maxima or minima.

In the exercise you are working with, the function \(g(x) = 3x^3 - 5\) is an example of a cubic function. For this specific function:

• The coefficient of \(x^3\) is 3, which affects the steepness of the curve.
• The constant \(-5\) shifts the graph vertically downward by 5 units.

Cubic functions like this one are continuous and smooth, without breaks or sharp corners, making them easily identifiable in graphs.

Graphing Functions

Graphing functions allows us to visually understand the behavior of functions. When graphing a cubic function like \(g(x) = 3x^3 - 5\), you will observe the distinctive "S" shaped curve.To graph such functions:

• Identify key points like the origin, where the function crosses the axes.
• Look for symmetry, which may not be as apparent in cubic functions compared to quadratics.
• Check the direction of the curve's increase or decrease, which depends on the sign of the highest coefficient.

Graphing both a function and its inverse on the same axes provides insights into their relationships. When plotting \(g(x)\) and its inverse \(g^{-1}(x)\), you'll notice that they will mirror each other across the line \(y = x\).For additional clarity:

• Use a solid line for the original function and a dashed line for the inverse to differentiate between them easily.
• Label each function on the graph to avoid confusion.

Function Symmetry

Symmetry in functions provides significant insights into their behaviors. The most relevant type of symmetry for inverse functions is reflection symmetry over the line \(y = x\). When a function and its inverse are graphed, they are symmetrical around this line.Here’s why symmetry is crucial:

• It highlights the visual relationship between functions and their inverses. They appear as mirror images across \(y = x\).
• This property helps verify if you've correctly calculated the inverse.

In our example, by finding the inverse of \(g(x) = 3x^3 - 5\):

• We plotted both \(g(x)\) and its inverse \(g^{-1}(x) = \sqrt[3]{(x+5)/3}\).
• Both should appear as mirror reflections across the line \(y = x\).

Observing this symmetry is crucial as it visually confirms the relationship between the function and its inverse, aligning with the mathematical calculations.