Pregunta
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
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-
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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
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
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:
-
for all in the domain of . -
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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