Problem 62 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine algebraically whether the function is one-to-one. Verify your answer graphically. If the function is one-to-one, find its inverse. $$f(x)=3 x+5$$

Step-by-Step Solution

Answer

The function \(f(x) = 3x + 5\) is one-to-one, and its inverse is \(f^{-1}(x) = (x - 5)/3\).

Step 1: Determine if the function is one-to-one

To check if \(f(x)\) is one-to-one, each x-value in the function must correspond to a unique y-value. A function \(f(x)\) is one-to-one if \(f(a) \neq f(b)\) for any \(a \neq b\). As there are no repeated values in \(3x+5\), this function is one-to-one.

Step 2: Graphical Verification

A graph of \(f(x) = 3x + 5\) is a straight line with slope 3 and y-intercept 5. This is a rising graph with no recurring y-values, supporting that \(f(x)\) is one-to-one. As the Horizontal Line Test (any horizontal line touches the graph at most once) is satisfied, the function is indeed one-to-one.

Step 3: Find the inverse of the function

If \(f(x)\) is one-to-one, it has an inverse. To find \(f^{-1}(x)\), swap x and y in the equation and solve for y. Substitute \(f(x)\) with y to get \(y = 3x + 5\). Swap x and y to get \(x = 3y + 5\). Solve for y to get the inverse function \(f^{-1}(x) = (x - 5)/3\).

Key Concepts

Inverse Functions

In mathematics, inverse functions are pairs of functions that 'undo' each other. For a function to have an inverse, it must be one-to-one; that means every output (y-value) is produced by exactly one input (x-value). To navigate between a function and its inverse, you essentially swap the roles of the input and output.

For instance, if you have a function like the textbook exercise's function, \( f(x) = 3x + 5 \), to find its inverse, you begin by replacing f(x) with y, making the equation \( y = 3x + 5 \)}. Then, exchange x and y and solve for the new y. You'll get the inverse function: \( f^{-1}(x) = \frac{x - 5}{3} \)}. The process of finding an inverse requires the function to be bijective, meaning both injective (one-to-one) and surjective (onto). This ensures every element in the target set is covered and that the inverse is a function as well.

Horizontal Line Test

A reliable way to visually confirm if a function is one-to-one is the Horizontal Line Test. This method involves drawing horizontal lines across a function's graph on a coordinate plane. If any horizontal line intersects the graph at more than one point, the function is not one-to-one.

Take our textbook function as an example. The graph of \( f(x) = 3x + 5 \)} is a straight, upward-sloping line. Testing it with horizontal lines drawn at various y-values reveals that no line will touch the graph more than once. That's proof it's one-to-one! It's important for students to understand that this test is just a graphical affirmation of a function's injectivity, affirming that the algebraic steps taken to find an inverse will be valid.

Function Graph Analysis

When analyzing the graph of a function like in our exercise, understanding its shape, direction, and intersection points is fundamental. For the function \( f(x) = 3x + 5 \)}, we observe that its graph is a line, indicating it's a linear function. It has a slope (rise over run) of 3, making it steep and confirming a constant rate of increase. The y-intercept is at (0, 5), revealing where the line crosses the y-axis.

In-Depth Look at Key Features

By examining these key features, we comprehend the function's behavior without plugging in specific x-values. The positive slope shows continuous growth; as x increases, so does y, and the steeper the slope, the faster y grows. These insights support the one-to-one nature of the function because a straight line doesn't double back on itself—it ensures a single y-value for every x-value, thereby validating the function's candidacy for an inverse.