4.1: Inverse Functions
Contents
Objectives:
- Use the horizontal line test to determine if a function is invertible
- Find the explicit formula for the inverse and give interpretations of the inverse function
Important Items
Definitions:
invertible, inverse function
Notes:
In this lesson, we will NOT be teaching students to sketch an inverse function by reflecting over the line Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=x} or to verify that a function is an inverse function by using the fact that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(f^{-1}(x))=x} . This notion does come up on problem 7 but you can have them discover this on their own.
Also, in high school a lot of students will have learned a procedure similar to the following for finding a formula for the inverse of a function:
- Start with the formula of the function,
- Swap the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} 's and the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} 's in this formula.,
- Manipulate to make Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} the subject, i.e. "solve for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} in the new formula".
The advantage of swapping is that students are more accustomed to solving for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} and so are less likely to become confused as to what algebraic manipulations are needed in Step 3. The (huge) disadvantage is that if the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} actually stand for real-world quantities then you also have to swap the meaning of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} when you calculate the formula for the inverse. This is a tough point for a lot of students. Namely, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is in dollars and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} is in apples in the original function, then they expect Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} to be in dollars and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} to be in apples in the inverse as well. It is for this reason that we advocate an approach to computing inverses in which you simply re-arrange an equation. Hence you must be careful to imitate follow the examples given here.
Lesson Guide
Warm-Up
Have students do Problems 1 and 2.
Use the horizontal line test to determine if a function is invertible
We say a function is invertible if its inverse is also a function.
Remind students the definition of a function.
A function is a relation (or a rule) that assigns each input to only one output.
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Suggested Example |
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This is a sentence that is in place to generate space in the table |
Walk students through problem 3 as an example. This is a very helpful way to teach students how to find the explicit formula of the inverse function because it emphasizes the order. Using the bubble diagram is also useful in other areas of the course. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = 3\sqrt{x+1}-4} . When we are computing what happens to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} , we first add 1 to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} , then take the square root, and so on. a) Represent this chain of computations visually by labeling the arrows in the diagram below.
https://www.desmos.com/calculator/x9qlb082jn b-c) Represent this chain of computations visually by labeling the arrows in the diagram below.
https://www.desmos.com/calculator/jcprbs1x4m d) Allow students to try to find the inverse of the invertible functions in Problem 3. The next objective will delve into this more deeply, and students may need to go back to these problems after seeing Objective (ii) in order to finish them.
At the end, introduce the horizontal line test as a way to verify whether or not a graph represents a function. Talk about how this is connected to the vertical line test and the graph of a function. It may be helpful to talk about the graph of the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = 3\sqrt{x+1}-4} .
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Find the explicit formula for the inverse and give interpretations of the inverse function
Have students do Problem 4.
Given a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P=f(Q),}
the notation for the inverse function will be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q=f^{-1}(P).}
Emphasize to your students that the inverse notation does not mean ``reciprocal." Furthermore, make sure your students understand that the inputs and outputs, or the domain and range, of an invertible function become the outputs and inputs, or the range and domain, respectively, of the inverse function.
Have students do Problems 5-8.
Comments
- I found it productive to use the second warm-up problem to transition into what an invertible function. We talked about it as a class first, put up an example, and then I defined invertible functions.
- It may be good to have an example of an example that interprets the meaning of an inverse for an exponential function, as a way to motive 4.2: Logarithms. You could end the example by saying something like: "And now next time, we'll figure out how to find the inverse of an exponential explicitly."
- Note: Problems 6 and 8 present functions that are only invertible on restricted domains (Problem 8 gives the restricted domain). I think it would be helpful to scaffold this idea a little bit more than what is present in the lesson plan. For example, with Problem 6 part (b) I think it'd be helpful to have a whole class discussion about why it makes sense not to include the negative root (i.e., we don't want to have negative values for "t" because we only know what the population is after 1982, not before), and so in fact this function can be made invertible if we restrict our domain to nonnegative values for time.
- A HEAD'S UP: I wish I had paid more attention to the bubble diagrams/emphasized them a little bit more when I taught this course. The diagrams pop up again in 5.1 (Compositions), and can be useful for explaining why the order of transformations makes sense (5.6). Although, if you haven't been stressing these sorts of diagrams by the time your class gets to 5.6, it can feel a little abrupt to start heavily emphasizing them in that section. TLDR: plan accordingly, make sure to look ahead to the lesson plan for 5.6.
