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4.1 - Verifying Invers…
4.1 - Verifying Inverses (Linear and Cubic)
Use the theorem on inverse functions to pro
and sketch the graphs of f and g on the same

Ask by Whittaker Gibbs. in the United States
Jan 23,2025

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All four pairs of functions and are inverses of each other. Their graphs are reflections of each other across the line .

Solución

Sure! Let’s verify whether each pair of functions and are inverses of each other using the theorem on inverse functions. We’ll also discuss how their graphs relate graphically.

Theorem on Inverse Functions

Two functions and are inverses of each other if and only if:
  1. for all in the domain of .
  2. for all in the domain of .
When two functions are inverses, their graphs are reflections of each other across the line .
Let’s apply this to each pair of functions provided.

1. and

Verification:

  • Calculate :
  • Calculate :
Since both compositions yield , is indeed the inverse of .

Graphical Interpretation:

  • The graph of is a straight line with slope and y-intercept at .
  • The graph of is a straight line with slope and y-intercept at .
  • These lines are reflections of each other across the line .

2. and

Verification:

  • Calculate :
  • Calculate :
Both compositions yield , so is the inverse of .

Graphical Interpretation:

  • The graph of is the cube root function shifted down by 1 unit.
  • The graph of is the cubic function shifted left by 1 unit.
  • These graphs are reflections of each other across the line .

3. and

Verification:

  • Calculate :
  • Calculate :
Since both compositions equal , is the inverse of .

Graphical Interpretation:

  • The graph of is a straight line with slope and y-intercept at .
  • The graph of is a straight line with slope and y-intercept at .
  • These lines are reflections of each other across the line .

4. and

Verification:

  • Calculate :
  • Calculate :
Both compositions result in , confirming that is the inverse of .

Graphical Interpretation:

  • The graph of is the cube root function shifted left by 2 units.
  • The graph of is the cubic function shifted down by 2 units.
  • These graphs are reflections of each other across the line .

Summary of Graphs

For each pair of functions and :
  • Verification: We showed and , proving that is the inverse of .
  • Graphical Relationship: The graphs of each pair and are mirror images across the line , illustrating their inverse relationship.
If you’d like to see the graphs plotted, you can use graphing software or tools like Desmos to visualize these functions and their inverses.

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Bonus Knowledge

To verify that two functions are inverses, check if and for each pair. For instance, with and , if you substitute into , you’ll find that demonstrating they are indeed inverses. Sketching the graphs will reveal that they are reflections across the line , showcasing their inverse relationship beautifully!
Have fun experimenting with transformations! For and , applying the cube function “undoes” the cube root, resulting in the identity function. When graphing these, make sure to plot points around critical values, such as the origin and any intercepts. You’ll see how each function complements the other, creating an intriguing symmetry!

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