Question

Find the inverse of the function. Then graph the function and its inverse. $$ y=3 x+5 $$

Short answer

The inverse of the function \( y = 3x + 5 \) is \( y = (x - 5) / 3 \). The graph of this function and its inverse reveals them to be reflections of each other over the line \( y = x \).

Step-by-step solution

Swap x and y

The first step in finding the inverse of a function is to swap the roles of x and y. This gives the equation \( x = 3y + 5 \).

Solve for y

Next, rearrange the equation by solving for y to isolate the inverse function. So, \( y = (x - 5) / 3 \).

Graph the functions

Plot both the original function \( y = 3x + 5 \) and its inverse \( y = (x - 5) / 3 \) on the same graph. Clearly, the inverse is a reflection of the original function over the line \( y = x \).

Key concepts

Understanding Inverse Functions

When we talk about inverse functions, we're looking at two functions that essentially 'undo' each other. In mathematical terms, if you have a function that takes an input 'x' and gives an output 'y', the inverse function would take 'y' as an input and give you back 'x'. To find the inverse of a function, you'll often start by switching the places of 'x' and 'y' in the equation and then solving for 'y'. This switcheroo is a crucial first move because it repositions your output as the new input, preparing for the role reversal.

Following this, it's all about getting 'y' alone on one side of the equation to have a proper function. For example, in our exercise, we started with the linear function \( y = 3x + 5 \) and swapped to get \( x = 3y + 5 \), which we then rearranged to get the inverse function \( y = (x - 5) / 3 \). Another crucial aspect of inverse functions is that not all functions have inverses. A function must be bijective, meaning it should be both injection and surjection, to have an inverse that is also a function.

In simpler words, for each 'x' there should be one and only one 'y', and for each 'y' one and only one 'x'; this is known as the 'one-to-one' property. It ensures that the inverse will accurately map back to the original 'x' without any ambiguity. If you're wondering about a quick check for this property, you can use the Horizontal Line Test on the graph of the function. If any horizontal line intersects the graph at more than one point, the function doesn't pass the test, and its inverse won't be a function.

Graphing Linear Functions

When it comes to graphing linear functions, the process is quite straightforward because linear functions form straight lines. These functions have the general form of \( y = mx + b \) where 'm' represents the slope, and 'b' is the y-intercept. The slope dictates how steep the line is, and the y-intercept tells us where the line crosses the Y-axis.

To graph a linear function, you can start by plotting the y-intercept on the Cartesian plane. From there, use the slope to find another point: move up or down depending on whether the slope is positive or negative and then move right to complete the triangle that the slope represents. Once you have two points, you can draw a line through them, and there you have it, the graph of the function.

In our original function \( y = 3x + 5 \), the slope is 3 (which means you go up three units for every one unit you go to the right), and the y-intercept is 5 (where the line crosses the Y-axis). If you want to see the inverse function on the same graph, you'd do the same thing with its slope and y-intercept. However, since the inverse function has the form \( y = (x - 5) / 3 \), its slope is \( 1/3 \), and its y-intercept is \( -5/3 \), after simplifying.

Reflecting Over the Line y=x

The 'landmark' for understanding the relationship between a function and its inverse on a graph is the line \( y = x \). If you reflect the graph of a function over this line, you will get the graph of its inverse. This is a visual manifestation of the swap we did with our 'x' and 'y' in the very first step of finding the inverse. The line \( y = x \) acts like a mirror, and the function and its inverse are mirror images of each other.

The significance of the line \( y = x \) is that it's where the input and output are the same (every point on this line is \( (a, a) \), meaning the x-coordinate and y-coordinate are identical). So, when you perform the reflection, the input of the original function becomes the output of the inverse function, and vice versa.

Visualizing the Reflection

It helps to actually draw the line \( y = x \) on your graph paper when you're plotting the function and its inverse. You'll notice that every point on the original function will have a 'partner' point on the inverse function that's located diagonally opposite across the line \( y = x \). This reciprocal nature is essential to understand because it enforces the concept that functions and their inverses are twofold ways of looking at the same relationship between variables.