Problem 32 - Algebra 2

Question

For Exercises \(31-34, f(x)=10 x-10 .\) Find each value. $$ \left(f \circ f^{-1}\right)(-10) $$

Step-by-Step Solution

Answer

The value of \((f \circ f^{-1})(-10)\) is -10.

Step 1: Understand the provided function f(x)

The function given in the exercise is \(f(x)= 10x-10\). This is a linear function.

Step 2: Understand the composition \(f \circ f^{-1}\)

The composition \(f \circ f^{-1}\) means that we are applying the function \(f\) to its inverse. This composition will always yield the original input, as per the properties of inverse functions. In other words, for any value \(x\), we have \(f(f^{-1}(x))=x\).

Step 3: Apply the value to the composition

Now we apply the value -10 to this composition. As we noted in the previous step, \((f \circ f^{-1})(-10) = -10\), because the composition \(f(f^{-1}(x))\) always yields the original input.

Key Concepts

Composition of Functions

When we talk about the composition of functions, we are essentially dealing with applying one function to the result of another. This is a foundational concept in algebra and calculus. The notation \((f \circ g)(x)\) can be read as "\(f\) of \(g\) of \(x\)," which means that you first apply \(g\) to \(x\), and then apply \(f\) to the result.

• The order of operations is crucial in function composition. This means if you have \((f \circ g)(x)\), it is not the same as \((g \circ f)(x)\).
• The composition of functions builds on the concept of substitution, where one function is "substituted" into another.

Understanding function composition aids in working through complex equations and facilitates solving equations with multiple operational layers.

Linear Functions

Linear functions are one of the simplest types of functions and are hence quite fundamental in mathematics. They are characterized by a constant rate of change, which geometrically is represented by a straight line. The standard form for a linear function is \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

• The slope \(m\) indicates how steep the line is. A positive \(m\) means the line goes upwards, while a negative \(m\) means it goes downwards.
• The y-intercept \(b\) is the point where the line crosses the y-axis.

Linear functions are extensively used to model real-world situations where a constant relationship exists between two variables. They are foundational in algebra and essential in calculus as a stepping stone to understanding more complex functions.

Properties of Inverse Functions

Inverse functions are special functions that essentially "reverse" the effect of the original function. If \(f\) is the original function and \(f^{-1}\) is its inverse, then applying them in succession will return you to the starting value. This is fundamental in solving equations and understanding transformations.

• The relationship \(f(f^{-1}(x)) = x\) shows that applying \(f^{-1}\) first and then \(f\) yields the original value \(x\).
• Conversely, \(f^{-1}(f(x)) = x\) demonstrates that applying \(f\) first and then \(f^{-1}\) will also result in \(x\).

These properties are not only vital in algebra but are also critical in calculus, especially when dealing with derivatives and integrals. They provide a deeper understanding of how functions and their inverses interact.