Problem 38 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Determine whether the inverse of \(f\) is a function. Then find the inverse. $$f(x)=\frac{7}{x+6}$$

Step-by-Step Solution

Answer

The inverse of the function \(f(x) = \frac{7}{x+6}\) is \(f^{-1}(x) = \frac{7}{x} - 6\). This inverse is a function for all real x values except when x = 0.

Step 1: Finding the Inverse

Switch x and y in the function and solve for y. \\ Begin by replacing \(f(x)\) with \(y\) to get \(y = \frac{7}{x+6}\). Then swap x and y to get \(x = \frac{7}{y+6}\). To isolate y in the equation, multiply both sides by \(y+6\) and then subtract 6 from both sides to get \(y = \frac{7}{x} - 6\). This is the inverse function, denoted by \(f^{-1}(x) = \frac{7}{x} - 6\)

Step 2: Determine if the Inverse is a Function

The existence and uniqueness of the y value of the inverse function can be checked by the vertical line test. In this case, the inverse passes the vertical line test for all real x except x = 0 (as it would make the denominator zero and the function undefined). So, the inverse \(f^{-1}(x) = \frac{7}{x} - 6\) is a function when x ≠ 0.

Key Concepts

Understanding the Vertical Line Test

The vertical line test is a simple method to determine if a graph represents a function. If you can draw a vertical line anywhere on the graph that intersects the graph more than once, the graph does not represent a function.

Why? Because a function, by definition, assigns exactly one output (or y-value) to each input (x-value). If a vertical line crosses twice, that means an input is paired with more than one output, violating the rule of functions.

For an inverse to be a function, it must pass this test. In the case of the inverse function \(f^{-1}(x) = \frac{7}{x} - 6\), we observe that as long as \(x eq 0\), a vertical line will touch it only once. However, at \(x = 0\), the function is undefined due to division by zero. Thus, the inverse is a function only for \(x eq 0\).

Solving Equations to Find the Inverse

To find an inverse function, we first swap the dependent and independent variables—essentially switching roles between x and y. This involves several key steps:

• First, replace the function notation, like \(f(x)\), with \(y\). For example, rewrite \(f(x) = \frac{7}{x+6}\) as \(y = \frac{7}{x+6}\).
• Next, swap x and y to set up the equation \(x = \frac{7}{y+6}\).
• Finally, solve for y. This usually means isolating y on one side of the equation. Multiply both sides by \(y + 6\) and rearrange to isolate y, resulting in \(y = \frac{7}{x} - 6\).

This new equation is the inverse function, \(f^{-1}(x)\). Solving equations in this context is about careful algebraic manipulation to "reverse" the roles of input and output.

Exploring Function Notation

Function notation is a concise way to express relationships between variables. It includes the function name and the variable it acts upon. For example, \(f(x)\) denotes a function named f that takes \(x\) as the input.

This notation not only shows the operation being performed but also clarifies that \(x\) is the variable we can substitute with specific numbers.

When finding an inverse, function notation changes accordingly. The inverse is represented as \(f^{-1}(x)\), reading as "f inverse of x." This notation alerts us to the idea that the inverse operation "undoes" the original function.

Function notation makes it easier to conceptualize relationships and communicate solutions precisely—crucial when dealing with inverses and proving their functionality with tests like the vertical line test.