Problem 13 - Calculus of a Single Variable

Question

Use a graphing utility to graph the function. Then use the Horizontal Line Test to determine whether the function is one-to-one on its entire domain and therefore has an inverse function. \(f(x)=\frac{3}{4} x+6\)

Step-by-Step Solution

Answer

The function \(f(x) = \frac{3}{4}x + 6\) is one-to-one and hence has an inverse.

Step 1: Graph the Function

Using a graphing utility, plot the function \(f(x) = \frac{3}{4}x + 6\). This is a linear function with slope \(\frac{3}{4}\) and y-intercept at \(6\).

Step 2: Conduct Horizontal Line Test

The Horizontal Line Test can be used to determine if a function is one-to-one. To apply this test, a horizontal line is passed over the graph. If at any point the line intersects the graph more than once, the function is not one-to-one.

Step 3: Result of the Horizontal Line Test

Upon applying the Horizontal Line Test to the graph of our function, it can be seen that the test passes since any horizontal line only intersects the graph once. Thus, the function \(f(x) = \frac{3}{4}x + 6\) is one-to-one.

Step 4: Conclusion

Since the function is one-to-one, it implies that the function has an inverse.

Key Concepts

Inverse Function

An inverse function essentially reverses the effect of the original function. When we have a function \( f(x) \), its inverse, denoted as \( f^{-1}(x) \), swaps the roles of inputs and outputs. If you input \( b \) into \( f \) and get \( a \) (\( f(b) = a \)), then in the inverse function, inputting \( a \) returns \( b \) (\( f^{-1}(a) = b \)).
For a function to have an inverse, it must be one-to-one. This one-to-one nature ensures that each output is paired with precisely one input, making the "reversal" possible. In graphical terms, the function must pass the Horizontal Line Test. Passing this test means any horizontal line touches the graph at most once. Thus, one unique input for every output, allowing us to create the inverse function without any ambiguity.
In our exercise, we determined that \( f(x) = \frac{3}{4}x + 6 \) passes the Horizontal Line Test, indicating it's one-to-one and thus has an inverse.

One-to-One Function

A function is called one-to-one if different inputs map to different outputs. In other words, each element of the function’s range is mapped from one distinct element of its domain.
Mathematically, a function \( f \) is one-to-one if for any two values \( x_1 \) and \( x_2 \) in the domain of \( f \), whenever \( f(x_1) = f(x_2) \), it follows that \( x_1 = x_2 \).
This property is crucial for functions that require an inverse because it ensures no two different inputs are associated with the same output. This makes the reversal in inverses possible – making it clear how to revert to the original input from an output.The Horizontal Line Test is a quick visual tool to confirm a function's one-to-one nature. Concretely, passing this test tells us that for every output value (y-value), there is only one corresponding input value (x-value). Our function \( f(x) = \frac{3}{4}x + 6 \) is one-to-one as horizontal lines intersect its graph at most once.

Graphing Functions

Graphing functions is an effective way to visually analyze the behavior of a mathematical function. When you graph a function, you're essentially plotting all possible outputs \( y \) for inputs \( x \) over the function's domain. This makes it much easier to understand characteristics like slope, intercepts, continuity, and whether or not the function is one-to-one.
For linear functions like \( f(x) = \frac{3}{4}x + 6 \), they appear as straight lines on the graph. The constant slope (\( \frac{3}{4} \)), indicates the line's steepness and direction, while the y-intercept (\( 6 \)) tells us where the line crosses the y-axis.
Graphing utilities or calculators can automate and simplify this process, producing accurate visual representations of more complex functions. By closely viewing these graphical outputs, you can quickly perform the Horizontal Line Test to determine if the function is one-to-one, as discussed previously in our solution.