Question

If \(g(-5)=3\) and \(g\) is one-to-one, what is \(g^{-1}(3) ?\)

Short answer

g^{-1}(3) = -5.

Step-by-step solution

Understand the given information

We are given that the function value at g(-5) = 3 and that the function g is one-to-one (bijective). A one-to-one function means that each output is associated with exactly one input.

Define the inverse function

The inverse function, denoted as g^{-1}(x), reverses the original function. This means that if g(a) = b, then g^{-1}(b) = a.

Use the given information

From g(-5) = 3, we can infer that applying the inverse function to the output 3 will give us the input -5.

Find g^{-1}(3)

Since g(-5) = 3, it follows that g^{-1}(3) = -5.

Key concepts

one-to-one functions

A one-to-one function, also known as an injective function, is a type of function where each output value is paired with exactly one unique input value. This means that no two different inputs will produce the same output.
Understanding one-to-one functions is crucial when dealing with inverse functions, as only one-to-one functions have inverses that are also functions.
Here are some important points about one-to-one functions:

• If a function is one-to-one, you can always determine the specific input from any given output.
• An easy way to check if a function is one-to-one is to use the Horizontal Line Test. Draw horizontal lines through the graph of a function. If any horizontal line intersects the graph at more than one point, the function is not one-to-one.

In this exercise, since g is one-to-one, knowing that g(-5) = 3 lets us definitively say that no other input besides -5 will produce the output 3.

inverse function definition

An inverse function essentially reverses a given function. If you have a function denoted by g that transforms an input a to an output b, its inverse, denoted by g^{-1}, will take b as an input and return a.
In mathematical terms:

• If g(a) = b, then g^{-1}(b) = a.
• The notation g^{-1} is used to represent the inverse function.

This concept is particularly useful when you need to 'undo' the action of a function.
Since inverse functions reverse the roles of inputs and outputs, they provide a way to backward trace from results to their originating inputs.
In the given problem, we are given g(-5) = 3 and we need to find g^{-1}(3). According to the definition of inverse functions, if g(-5) = 3, then g^{-1}(3) must equal -5.

function evaluation

Function evaluation is the process of finding the output of a function given an input. For example, if you have a function g(x) and you're given an input value, evaluating the function means calculating g(some value). In the given problem, it is stated: g(-5) = 3, meaning that if we plug -5 into the function g, the output is 3.
When dealing with inverse functions, the roles are reversed: instead of plugging in an input to get an output, you plug in an output to find the original input.
Given g(-5) = 3, if we evaluate the inverse function g^{-1} at 3, we need to find which input for g produced 3 as an output. By definition of inverse functions, we know g^{-1}(3) = -5.
This reverse approach of function evaluation allows us to trace outputs back to their corresponding inputs, making it particularly useful in various types of mathematical problems.

• Evaluating g(-5) = 3 indicates simple forward function evaluation.
• Finding g^{-1}(3) requires reverse function evaluation using the inverse function.

In summary: given g(-5) = 3, we applied inverse function reasoning to find that g^{-1}(3) = -5.