Problem 44 - Intermediate Algebra

Question

(a) find the inverse of the given function, and (b) graph the given function and its inverse on the same set of axes. (Objective 4) $$f(x)=-6 x$$

Step-by-Step Solution

Answer

The inverse of the function is \( f^{-1}(x) = -\frac{1}{6}x \). Graph both \( f(x) = -6x \) and \( f^{-1}(x) \) to visualize their symmetry across \( y = x \).

Step 1: Identify the Function

The given function is the linear function \( f(x) = -6x \). This function is a straight line with a slope of -6.

Step 2: Express the Function in Terms of y

Replace \( f(x) \) with \( y \) to express the function as an equation: \( y = -6x \).

Step 3: Solve for the Inverse Function

To find the inverse function, switch \( x \) and \( y \) and solve for \( y \). Start by swapping to get \( x = -6y \), then solve for \( y \) by dividing both sides by -6. This gives \( y = -\frac{1}{6}x \). Thus, the inverse function is \( f^{-1}(x) = -\frac{1}{6}x \).

Step 4: Check the Inverse Function

Verify the inverse function by checking that composing \( f \) with \( f^{-1} \) gives the identity function: \( f(f^{-1}(x)) = -6(-\frac{1}{6}x) = x \), and \( f^{-1}(f(x)) = -\frac{1}{6}(-6x) = x \). Both compositions return \( x \), confirming the inverse is correct.

Step 5: Graph the Function and Its Inverse

Graph \( f(x) = -6x \) as a downward sloping line passing through the origin with a slope of -6. Then graph its inverse \( f^{-1}(x) = -\frac{1}{6}x \) as another line passing through the origin with a gentler slope of \(-\frac{1}{6}\). Both graphs will be symmetrical across the line \( y = x \).

Key Concepts

Linear Functions

Linear functions are the foundation for many algebraic concepts and are represented by the general formula \( y = mx + b \). In this equation, \( m \) is the slope, which indicates the steepness of the line, and \( b \) is the y-intercept, where the line crosses the y-axis. For the function \( f(x) = -6x \), this is a special case of a linear function as the intercept \( b \) is 0, making it a line through the origin.

The slope here is \(-6\), which means for every unit increase in \( x \), \( y \) decreases by 6 units. This negative slope results in a downward slope through the graph.

• Linear functions graph as straight lines.
• The direction of the slope indicates if the line is increasing or decreasing.
• A larger absolute value of the slope indicates a steeper line.

Remember, linear functions are straightforward as they have no exponents higher than 1 for \( x \), and their graphs will always be a straight line, providing a basic yet powerful tool for representing relationships in mathematics.

Composition of Functions

The composition of functions involves creating a new function by combining two functions, typically denoted as \( (f \circ g)(x) = f(g(x)) \). When working with inverses, function composition is useful to verify that one function is indeed the inverse of another.

For example, given \( f(x) = -6x \) and its inverse \( f^{-1}(x) = -\frac{1}{6}x \), we can check correctness by calculating \( f(f^{-1}(x)) \) and \( f^{-1}(f(x)) \):

• \( f(f^{-1}(x)) = -6(-\frac{1}{6}x) = x \)
• \( f^{-1}(f(x)) = -\frac{1}{6}(-6x) = x \)

Both calculations returning \( x \) confirms that the functions are inverses of each other.

This procedure might seem roundabout, but it's a crucial step in validating our mathematical solutions and understanding how functions can transform and interact with one another.

Function Graphing

Graphing functions, including their inverses, provides a visual representation of mathematical relationships and allows us to explore their properties easily. When graphing \( f(x) = -6x \) and its inverse \( f^{-1}(x) = -\frac{1}{6}x \), begin by plotting each as straight lines according to their slope values.

- **Graphing the Original Function**: - Plot a point at the origin (0,0), as both functions pass through this point. - For \( f(x) = -6x \), from the origin, move down 6 units and right 1 unit to place a second point and draw the line.- **Graphing the Inverse Function**: - Similarly, for \( f^{-1}(x) = -\frac{1}{6}x \), from the origin, move down 1 unit and right 6 units to place the second point and draw the line.These graphs should show symmetry across the line \( y = x \), which serves as a mirror for two functions that are inverses, reinforcing their mathematical relationship.

Understanding function graphing helps in clearly visualizing how functions behave, intersect, and transform across a coordinate plane.