Textbook Question
Use the definition of inverses to determine whether ƒ and g are inverses. f(x) = x2+3, x≥0; g(x) = √x-3, x≥3
595
views
3. Functions
Function Composition
1:01 minutesProblem 53
Textbook Question
Find the inverse of each function that is one-to-one. {(1, -3), (2, -7), (4, -3), (5, -5)}
Verified step by step guidance1
First, understand that the inverse of a function swaps the roles of inputs and outputs. For a function given as a set of points, the inverse will have each ordered pair reversed, meaning the x- and y-values switch places.
Check if the function is one-to-one by verifying that each output value (y-value) is unique. If any y-value repeats, the function is not one-to-one and does not have an inverse function.
Since the function is given as the set {(1, -3), (2, -7), (4, -3), (5, -5)}, observe that the output -3 appears twice (for inputs 1 and 4), so the function is not one-to-one and does not have an inverse function over the entire domain.
If you were to find the inverse for a one-to-one function, you would create a new set of ordered pairs by swapping each (x, y) to (y, x). For example, if the function had pairs (a, b), the inverse would have pairs (b, a).
In summary, since the function is not one-to-one, it does not have an inverse function. However, if restricted to a domain where outputs are unique, you could find the inverse by swapping coordinates as described.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
One-to-One Functions
A function is one-to-one if each output corresponds to exactly one input, meaning no two different inputs share the same output. This property is essential for a function to have an inverse because the inverse must assign a unique output to each input.
Recommended video:
4:07
Decomposition of Functions
Inverse Functions
The inverse of a function reverses the roles of inputs and outputs, mapping each output back to its original input. For a function to have an inverse, it must be one-to-one, ensuring the inverse is also a function.
Recommended video:
4:30Graphing Logarithmic Functions
Function Representation Using Ordered Pairs
Functions can be represented as sets of ordered pairs (input, output). Finding the inverse involves swapping each pair to (output, input), but only if the original function is one-to-one to maintain the definition of a function.
Recommended video:
5:31Graphing Rational Functions Using Transformations
Related Videos
Related Practice