Problem 76 - Elementary and Intermediate Algebra

Question

Write in words how to read the notation. a. \(f^{-1}(x)=\frac{1}{2} x-3\) b. \(\left(f \circ f^{-1}\right)(x)=x\)

Step-by-Step Solution

Answer

a. "f inverse of x equals one half x minus three." b. "The composition of f and f inverse of x equals x."

Step 1: Identifying the Inverse Function

In part (a), we have the notation \( f^{-1}(x) = \frac{1}{2}x - 3 \). This is read as "f inverse of x equals one half x minus three." Here, \( f^{-1}(x) \) is the inverse function of \( f(x) \). It represents the operation that will reverse the action of the function \( f(x) \).

Step 2: Reading Function Composition

In part (b), the notation \( \left(f \circ f^{-1}\right)(x) = x \) requires reading the composition of functions. This is read as "the composition of f and f inverse of x equals x." The symbol \( \circ \) represents the composition of functions. In this context, the composition of \( f(x) \) and its inverse \( f^{-1}(x) \) gives the identity function, which returns \( x \).

Key Concepts

Function Notation

Function notation is a way to represent mathematical functions in a concise and standardized form. It allows you to easily identify the function and its related operations.
Instead of writing out the complete words describing a function, we use symbols and variables to express these operations.
Function notation is typically written as \( f(x) \), where \( f \) is a symbol representing the function name, and \( x \) is the variable input into the function.
For example, for the expression \( f(x) = 2x + 3 \), \( f(x) \) signifies that the output value, depending on the input \( x \), is computed by applying the formula \( 2x + 3 \).
An inverse function is represented by \( f^{-1}(x) \). The "inverse" here implies that it can reverse the effect of \( f(x) \).
This is crucial in various fields like algebra and calculus, where reversing operations can solve equations or simplifies computation.

Function Composition

Function composition involves creating a new function by combining two functions. This is symbolized by the composition operator \( \circ \).
When you see a composition notation \( (f \circ g)(x) \), it means you first apply the function \( g \) and then apply the function \( f \) to the result of \( g(x) \).
In other words, if you have \( g(x) \) and \( f(x) \), then \( (f \circ g)(x) = f(g(x)) \). This approach is useful to address complex functional problems by breaking them down into simpler, manageable operations.
For example, if \( f(x) = 2x \) and \( g(x) = x + 3 \), then \( (f \circ g)(x) = 2(x + 3) = 2x + 6 \).
When applying it to inverse scenarios, such as \( (f \circ f^{-1})(x) \), the composition results in the identity function.

Identity Function

The identity function is an important concept where the output is exactly the same as the input. Symbolically, it is represented as \( I(x) = x \).
It acts as a "do nothing" function, returning the input unchanged when passed through the function.
This function plays a crucial role when discussing inverses. When you compose a function with its inverse, such as \( (f \circ f^{-1})(x) \), it results in the identity function.
This is because the inverse effectively "undoes" the function's transformation, bringing you back to the original input \( x \).
Understanding identity functions is vital for solving equations and understanding more complex mathematical frameworks, simplifying problems to determine outcomes directly proportional to the inputs.