Question

In Exercises \(13-16,\) use the Horizontal Line Test to determine whether the function is one-to-one on its entire domain and therefore has an inverse function. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. $$ f(x)=\frac{3}{4} x+6 $$

Short answer

The function \(f(x) = \frac{3}{4}x + 6\) is a linear function, and as such, it is one-to-one on its entire domain and has an inverse function. Any horizontal line drawn will always intersect the function at most once.

Step-by-step solution

The Horizontal Line Test (HLT)

Apply the Horizontal Line Test by drawing various horizontal lines across the graph of the function \(f(x) = \frac{3}{4}x + 6\). Each of these lines should intersect the graph at only one point

Check for intersections

Check if any of the horizontal lines drawn in Step 1 intersect the function more than once. If any line does, the function is not one-to-one.

Determine if the function has an inverse

If in step 2, none of the horizontal lines drawn intersect the function at more than one point, we say that the function passed the Horizontal Line Test and therefore is one-to-one. Being one-to-one implies an inverse function exists on the entire domain of the function.

Key concepts

Understanding One-to-One Functions

The concept of a one-to-one function is crucial in determining the uniqueness of relationships between two sets of values. In simpler terms, a one-to-one function, also known as an injective function, ensures that each input corresponds to exactly one unique output. This means that no two different inputs can have the same output value.

When visualizing this on a graph, any horizontal line should cut the function at most once, which is known as the Horizontal Line Test. If a horizontal line intersects the graph of the function at more than one point, then the function is not one-to-one. Let's apply this principle to the given exercise.

Applying the Horizontal Line Test

For the function in the exercise, it is provided that f(x) = \(\frac{3}{4}x + 6\). This function is a straight line with a constant slope, which means that any horizontal line we draw will only intersect the graph at one point. Thus, it passes the Horizontal Line Test, indicating it is a one-to-one function.

Decoding Inverse Functions

An inverse function essentially reverses the process of the original function. When a function is one-to-one, it guarantees that for every output there is a unique input. This allows us to 'flip' the function around the line y = x, giving us a mirror image that represents the inverse.

The inverse function undoes the operation of the original function. For example, if you have a function that adds 4 to any number, its inverse will subtract 4.

Finding the Inverse

In the context of the function from our exercise, f(x) = \(\frac{3}{4}x + 6\), the inverse would be found by rearranging the equation to solve for 'x' in terms of 'y'. Since this linear function is one-to-one, it definitely has an inverse. Given its simplicity, we know without even doing the full reversal process that an inverse exists.

Function Domain Explored

The domain of a function is the complete set of possible input values (usually 'x' values). It specifies the range of independent values for which the function is defined. In many functions, like the one in our exercise, the domain is all real numbers because you can plug any real number into f(x) = \(\frac{3}{4}x + 6\) and get out a real number.

However, it's important to know that not all functions have such a broad domain. Some functions have restrictions, such as not allowing the input to create an undefined expression like division by zero.

Checking the Domain

When analyzing the domain, look for any restrictions on 'x'. For linear functions like the one in our example, there are typically no restrictions, but more complex functions might require more exploration into their domain to understand where they are well-defined.