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Questions and Answers
If f(g(x)) and g(f(x)) both equal x, then the function g is the _____ function of the function f.
If f(g(x)) and g(f(x)) both equal x, then the function g is the _____ function of the function f.
inverse
Find the inverse function of f informally. $f(x) = 3x + 1$
Find the inverse function of f informally. $f(x) = 3x + 1$
$f^{-1}(x) = \frac{x-1}{3}$
Find the inverse function of f informally. $f(x) = \frac{x-3}{9}$
Find the inverse function of f informally. $f(x) = \frac{x-3}{9}$
$f^{-1}(x) = 9x+3$
Use the graph of the function to sketch the graph of its inverse function $y=f^{-1}(x)$.
Use the graph of the function to sketch the graph of its inverse function $y=f^{-1}(x)$.
Use the table of values for y = f(x) to complete a table for y = f⁻¹(x).
Use the table of values for y = f(x) to complete a table for y = f⁻¹(x).
Does the function have an inverse function?
Does the function have an inverse function?
Use a graphing utility to graph the function. Use the Horizontal Line Test to determine whether the function has an inverse function. $g(x) = (x + 2)^2 + 3$
Use a graphing utility to graph the function. Use the Horizontal Line Test to determine whether the function has an inverse function. $g(x) = (x + 2)^2 + 3$
Use a graphing utility to graph the function. Use the Horizontal Line Test to determine whether the function has an inverse function. $f(x) = (x + 3)^3$
Use a graphing utility to graph the function. Use the Horizontal Line Test to determine whether the function has an inverse function. $f(x) = (x + 3)^3$
Use a graphing utility to graph the function. Use the Horizontal Line Test to determine whether the function has an inverse function. $h(x) = x - \frac{6}{x+1} - 31$
Use a graphing utility to graph the function. Use the Horizontal Line Test to determine whether the function has an inverse function. $h(x) = x - \frac{6}{x+1} - 31$
Consider the following function. $f(x) = x^5 - 2$. Find the inverse function of f.
Consider the following function. $f(x) = x^5 - 2$. Find the inverse function of f.
Consider the following function. $f(x) = x^5 - 2$. Graph both f and f⁻¹ on the same set of coordinate axes.
Consider the following function. $f(x) = x^5 - 2$. Graph both f and f⁻¹ on the same set of coordinate axes.
Consider the following function. $f(x) = x^5 - 2$. Describe the relationship between the graphs of f and f⁻¹.
Consider the following function. $f(x) = x^5 - 2$. Describe the relationship between the graphs of f and f⁻¹.
Consider the following function. $f(x) = x^5 - 2$. State the domain and range of f. (Enter your answers using interval notation.)
Consider the following function. $f(x) = x^5 - 2$. State the domain and range of f. (Enter your answers using interval notation.)
Consider the following function. $f(x) = x^{3/5}$. State the domain and range of f⁻¹. (Enter your answers using interval notation.)
Consider the following function. $f(x) = x^{3/5}$. State the domain and range of f⁻¹. (Enter your answers using interval notation.)
Determine whether the function has an inverse function. $f(x) = x^6$
Determine whether the function has an inverse function. $f(x) = x^6$
If it does, then find the inverse function. (If an answer does not exist, enter DNE.) $f(x) = x^6$
If it does, then find the inverse function. (If an answer does not exist, enter DNE.) $f(x) = x^6$
Determine whether the function has an inverse function. $g(x) = \frac{x + 1}{9}$
Determine whether the function has an inverse function. $g(x) = \frac{x + 1}{9}$
If it does, then find the inverse function. (If an answer does not exist, enter DNE.) $g(x) = \frac{x + 1}{9}$
If it does, then find the inverse function. (If an answer does not exist, enter DNE.) $g(x) = \frac{x + 1}{9}$
Determine whether the function has an inverse function. $f(x) = \sqrt{x - 1}, x \ge 1$
Determine whether the function has an inverse function. $f(x) = \sqrt{x - 1}, x \ge 1$
If it does, then find the inverse function. (If an answer does not exist, enter DNE.) $f(x) = \sqrt{x - 1}, x \ge 1$
If it does, then find the inverse function. (If an answer does not exist, enter DNE.) $f(x) = \sqrt{x - 1}, x \ge 1$
Determine whether the function has an inverse function. $f(x) = \frac{4x + 7}{7x + 5}$
Determine whether the function has an inverse function. $f(x) = \frac{4x + 7}{7x + 5}$
If it does, then find the inverse function. (If an answer does not exist, enter DNE.) $f(x) = \frac{4x + 7}{7x + 5}$
If it does, then find the inverse function. (If an answer does not exist, enter DNE.) $f(x) = \frac{4x + 7}{7x + 5}$
Determine whether the function has an inverse function. $f(x) = (x + 9)^2, x \ge -9$
Determine whether the function has an inverse function. $f(x) = (x + 9)^2, x \ge -9$
If it does, then find the inverse function. (If an answer does not exist, enter DNE.) $f(x) = (x + 9)^2, x \ge -9$
If it does, then find the inverse function. (If an answer does not exist, enter DNE.) $f(x) = (x + 9)^2, x \ge -9$
Use the functions f(x) = x + 5 and g(x) = 3x^6 to find each of the following functions. $f^{-1}(x) = $
Use the functions f(x) = x + 5 and g(x) = 3x^6 to find each of the following functions. $f^{-1}(x) = $
Use the functions f(x) = x + 5 and g(x) = 3x^6 to find each of the following functions. $(g^{-1} \omicron f^{-1})(x) = $
Use the functions f(x) = x + 5 and g(x) = 3x^6 to find each of the following functions. $(g^{-1} \omicron f^{-1})(x) = $
The cost C for a business to make personalized T-shirts is given by $C(x) = 7.50x + 1500$ where x represents the number of T-shirts.
The cost C for a business to make personalized T-shirts is given by $C(x) = 7.50x + 1500$ where x represents the number of T-shirts.
Explain what C(x) and C⁻¹(x) represent in the context of the problem.
Explain what C(x) and C⁻¹(x) represent in the context of the problem.
Flashcards
What is an inverse function?
What is an inverse function?
If f(g(x)) and g(f(x)) both equal x, then the function g is the inverse function of the function f.
Informal Inverse Function
Informal Inverse Function
To find the inverse of a function informally, reverse the operations. For example, if f(x) = 3x + 1, then f⁻¹(x) = (x - 1)/3.
Horizontal Line Test purpose?
Horizontal Line Test purpose?
Use the horizontal line test on the graph of a function to determine if it has an inverse function. If any horizontal line intersects the graph more than once, the function does not have an inverse.
Graph of Inverse Functions
Graph of Inverse Functions
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Finding an Inverse Function
Finding an Inverse Function
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Table of Values for Inverse function
Table of Values for Inverse function
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Function is One-to-One
Function is One-to-One
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Function is NOT One-to-One
Function is NOT One-to-One
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Domain and Range
Domain and Range
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Domain and Range of Inverse
Domain and Range of Inverse
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How to calculate Inverse functions
How to calculate Inverse functions
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Study Notes
- If f(g(x)) and g(f(x)) both equal x, then the function g is the inverse function of the function f.
Finding Inverse Functions Informally
- To find the inverse function of f(x) = 3x + 1, first determine the the inverse function and then, verify that f(f⁻¹(x)) = x.
- The inverse function of f(x) = 3x + 1 is f⁻¹(x) = (x-1)/3.
- f(f⁻¹(x)) = 3((x-1)/3) + 1 = x - 1 + 1 = x.
- To verify that f⁻¹(f(x)) = x, determine f⁻¹(f(x)) = ((3x + 1) - 1)/3 = 3x/3 = x.
- The inverse function of f(x) = (x-3)/9 is f⁻¹(x) = 9x + 3.
- f(f⁻¹(x)) = ((9x + 3) - 3) / 9 = 9x / 9 = x.
- f⁻¹(f(x)) = 9((x-3)/9) + 3 = x - 3 + 3 = x.
Using Graphs to Sketch Inverse Functions
- To sketch the graph of an inverse function y = f⁻¹(x), use the existing graph of the function.
Completing Tables for Inverse Functions
- Given a table of values for y = f(x), a table for y = f⁻¹(x) can be completed by swapping the x and y values.
- For instance, if f(x) has the points (-1, 5), (0, 7), (1, 9), (2, 11), (3, 13), and (4, 15), then f⁻¹(x) has the points (5, -1), (7, 0), (9, 1), (11, 2), (13, 3), and (15, 4).
- Likewise if f(x) has the points (-4, 5), (-3, 0), (-2, -5), (-1, -10), (0, -15), and (1, -20), then f⁻¹(x) has the points (5, -4), (0, -3), (-5, -2), (-10, -1), (-15, 0), and (-20, 1).
Determining Existence of Inverse Functions
- A function has an inverse function if it passes the horizontal line test.
Using Graphing Utilities and the Horizontal Line Test
- Graphing utilities and the Horizontal Line Test can help determine if a function has an inverse function.
- If a function is one-to-one (passes the horizontal line test), it has an inverse function.
- If a function is not one-to-one (does not pass the horizontal line test), it does not have an inverse function.
Finding Inverse Functions Algebraically
- The inverse function of f(x) = x⁵ - 2 is f⁻¹(x) = (x + 2)^(1/5).
- The graph of f⁻¹ is the reflection of f in the line y = x. -The domain and range of both f and f⁻¹ are (-∞, ∞).
- The inverse function of f(x) = x^(3/5) is f⁻¹(x) = x^(5/3).
- The graph of f⁻¹ is the reflection of f in the line y = x.
- The domain and range of both f and f⁻¹ are (-∞, ∞).
Determining if a Function has an Inverse
- f(x) = x⁶ does not have an inverse.
- g(x) = (x + 1)/9 has the inverse g⁻¹(x) = 9x - 1.
- f(x) = 5x + 8 has the inverse f⁻¹(x) = (x - 8)/5.
- f(x) = √(x - 1), x ≥ 1 , has the inverse f⁻¹(x) = x² + 1, x ≥ 0
- f(x) = (4x + 7) / (7x + 5) has the inverse f⁻¹(x) = (7 - 5x) / (7x - 4).
- f(x) = (x + 9)², x ≥ -9 has the inverse f⁻¹(x) = √x - 9, x ≥ 0.
Combining Functions to Find Inverses
- Given f(x) = x + 5 and g(x) = 3x⁶, then f⁻¹(x) = x - 5, g⁻¹(x) = ⁶√(x/3), and (g⁻¹ o f⁻¹)(x) = ⁶√((x+5)/3).
Real-World Applications of Inverse Functions
- For a business, the cost C to make personalized T-shirts is given by C(x) = 7.50x + 1500, where x represents the number of T-shirts.
- C(x) represents the total cost of making x units of T-shirts, and C⁻¹(x) represents the number of T-shirts that can be made for a given total cost.
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