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Building Inverses of Functions

Building Inverses of Functions. Lesson 5.5. Gloria and Keith are sharing their graphs for the same set of data. “I know my graph is right!” exclaims Gloria. “I’ve checked and rechecked it. Yours must be wrong, Keith.”

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Building Inverses of Functions

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  1. Building Inversesof Functions Lesson 5.5

  2. Gloria and Keith are sharing their graphs for the same set of data. • “I know my graph is right!” exclaims Gloria. “I’ve checked and rechecked it. Yours must be wrong, Keith.” • Keith disagrees. “I’ve entered these data into my calculator too, and I made sure I entered the correct numbers.” • Can you explain what is happening?

  3. The Inverse • In this investigation you will use graphs, tables, and equations to explore the inverses of several functions. • Graph the equation f(x) =6+3x on your calculator. Complete the table for this function.

  4. Because the inverse is obtained by switching the independent and dependent variables, you can find five points on the inverse of function f by swapping the x- and y-coordinates in the table. Complete the table for the inverse.

  5. Graph the five points you found in the last by creating a scatter plot. Describe the graph and write an equation for it. Graph your equation and verify that it passes through the points in the table from the last step.

  6. Repeat the previous steps for each of these functions. You may need to write more than one equation to describe the inverse.

  7. Study the graphs of functions and their inverses that you made. What observations can you make about the graphs of a function and its inverse? • You create the inverse by switching the x- and y-values of the points. How can you apply this idea to find the equation of the inverse from the original function? Verify that your method works by using it to find the equations for the inverses of functions f, g, and h.

  8. Example A • A 589 mi flight from Washington, D.C., to Chicago took 118 min. A flight of 1452 mi from Washington, D.C., to Denver took 222 min. Model this relationship both as (time, distance) data and as (distance, time) data. • If a flight from Washington, D.C., to Seattle takes 323 min, what is the distance traveled? • If the distance between Washington, D.C., and Miami is 910 mi, how long will it take to fly from one of these two cities to the other?

  9. If you know the time traveled and want to find the distance, then time is the independent variable, and the points known are (118, 589) and (222, 1452). • The slope is or approximately 8.3 mi/min. • Using the first point to write an equation in point-slope form, you get To find the distance between Washington, D.C., and Seattle, substitute 323 for the time, t:

  10. If you know distance and want to find time, then distance is the independent variable. The two points then are (589, 118) and (1452, 222). This makes the slope or approximately 0.12 min/mi. Using the first point again, the equation for time is

  11. To find the time of a flight from Washington, D.C. to Miami, substitute 910 for the distance, d.

  12. You can also use the first equation for distance and solve for t to get the second equation, for time.

  13. Find the composition of this function with its inverse. f(x)=4-3x

  14. When you take the composition of a function and its inverse, you get x. How does the graph of y = x relate to the graphs of a function and its inverse? Look carefully at the graphs below to see the relationship between a function and its inverse.

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