Questions and Answers
How do you determine if a graph represents a function?
Use the Vertical Line Test
How can you determine if the inverse of f(x) is a function?
Use the horizontal line test to determine if it is one-to-one.
Does y = x^2 have an inverse that is a function?
False
Does y = 3x + 2 have an inverse that is a function?
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Does y = |x| have an inverse that is a function?
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What is the notation for the inverse of f(x)?
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What is the inverse of the point (2, -5)?
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What is the inverse of the point (3, 3)?
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When given the graph of a function, in order to graph the inverse, what should you do?
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On a graph, f(x) and f^(-1)(x) will be symmetric over which line?
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How can you find the inverse of a function algebraically?
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Find the inverse of f(x) = 2x + 5.
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What is true about the domains and ranges of inverses?
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Complete the statement that verifies that f(x) and g(x) are inverses: f(g(x))=g(f(x))=___
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If f(x) = 3x + 2 and g(x) = x + 6, what is the first step in finding f(g(x))?
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If f(x) = x + 3 and g(x) = x - 3, are these inverse functions?
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If f(x) = (x + 3)/2 and g(x) = 2/(x + 3), are these inverse functions?
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If f(x) = x + 3 and g(x) = x - 3, find f(g(x)).
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If the inverse of a function is also a function, we say that it is...
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Is the function y = x^2 one-to-one?
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Is the function y = x^3 one-to-one?
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Is the function y = |x| one-to-one?
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Is the function y = sqrt(x) one-to-one?
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Is the function y = 1/x one-to-one?
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The vocab name for (f o g)(x) and (g o f)(x) is?
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Study Notes
Identifying Functions
- A graph represents a function if it passes the Vertical Line Test.
Inverses and One-to-One Functions
- An inverse function exists if the original function passes the Horizontal Line Test, indicating it is one-to-one.
Specific Functions and Their Inverses
- The function y = x² does not have an inverse that is a function.
- The function y = 3x + 2 has an inverse that is a function.
- The function y = |x| does not have an inverse that is a function.
Inverse Notation
- The notation for the inverse of f(x) is f^(-1)(x), provided it is a function.
Finding Inverses of Points
- The inverse of the point (2, -5) is (-5, 2).
- The inverse of the point (3, 3) is itself (3, 3).
Graphing Inverses
- To graph the inverse of a function, select key points, switch their coordinates, and then plot the new points.
- On a graph, f(x) and f^(-1)(x) are symmetric over the line y = x.
Algebraic Inversion Process
- To find the inverse of a function algebraically, switch x and y, then solve for y.
Example of Algebraic Inverse
- The inverse of the function f(x) = 2x + 5 is y = (x - 5)/2.
Domains and Ranges of Inverses
- The domain of f(x) is the range of f^(-1)(x).
Composition of Functions
- For two functions f(x) and g(x) to be inverses, they must satisfy f(g(x)) = g(f(x)) = x.
Steps in Function Composition
- For functions f(x) = 3x + 2 and g(x) = x + 6, the first step in finding f(g(x)) is 3(x + 6) + 2.
Verifying Inverse Functions
- The functions f(x) = x + 3 and g(x) = x - 3 are inverses.
- The functions f(x) = (x + 3)/2 and g(x) = 2/(x + 3) are not inverses.
Evaluation of Functions
- If f(x) = x + 3 and g(x) = x - 3, then f(g(x)) = x.
- An inverse function is considered one-to-one if it itself is a function.
One-to-One Functions
- The function y = x² is not one-to-one.
- The function y = x³ is one-to-one.
- The function y = |x| is not one-to-one.
- The function y = sqrt(x) is one-to-one but requires a domain restriction for quadratic functions.
- The function y = 1/x is one-to-one.
Composition of Functions
- The term for (f o g)(x) and (g o f)(x) is known as Composition of Functions.
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Test your understanding of inverse functions with these flashcards. Learn how to determine if a graph represents a function and if its inverse is also a function. Master concepts such as the Vertical and Horizontal Line Tests.
