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Magazine Chapter 5: Problem 49
(a) Prove that the function defined by \(f(x)=a x+b\) (a linear function) for \(a \neq 0\) has an inverse function, and find \(f^{-1}(x)\). (b) Does a constant function have an inverse? Explain.
Short Answer
Expert verified
(a) \( f^{-1}(x) = \frac{x - b}{a} \). (b) No, a constant function is not one-to-one.
Step by step solution
01
Understand Linear Functions
A linear function is of the form \( f(x) = ax + b \), where \( a \) is the slope and \( b \) is the y-intercept. For the function to have an inverse, it must be bijective (one-to-one and onto). The condition \( a eq 0 \) ensures the function is one-to-one, as it creates a non-horizontal line with a constant slope.
02
Verify Bijectiveness of Linear Function
Since \( a eq 0 \), every input \( x \) corresponds to a unique output \( f(x) \), making the function one-to-one. Additionally, since the line extends infinitely in both directions, the function is onto the real numbers. A function that is both one-to-one and onto is bijective, so \( f(x) \) has an inverse.
03
Find the Inverse Function
To find the inverse, switch \( x \) and \( y \) in the equation \( y = ax + b \), giving us \( x = ay + b \). Solve for \( y \):\[ y = \frac{x - b}{a} \]. Thus, the inverse function \( f^{-1}(x) \) is \( f^{-1}(x) = \frac{x - b}{a} \).
04
Analyze Constant Functions
A constant function is of the form \( f(x) = c \), where \( c \) is a constant. For it to have an inverse, it must be one-to-one, meaning each output corresponds uniquely to an input. However, a constant function maps all inputs to the same output, so it is not one-to-one and does not have an inverse.
05
Conclusion Based on Analysis
The linear function \( f(x) = ax + b \) with \( a eq 0 \) has an inverse, which is \( f^{-1}(x) = \frac{x - b}{a} \). In contrast, a constant function \( f(x) = c \) does not have an inverse, as it is not bijective.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Linear Functions
Linear functions have a simple yet powerful form: \[ f(x) = ax + b \] where \( a \) is the slope and \( b \) is the y-intercept. These functions create straight lines when graphed. They can rise or fall with a constant rate, depending on the sign and value of \( a \).
When \( a eq 0 \), the function maintains a clear one-to-one relationship between inputs and outputs, making it eligible to have an inverse.
- The slope \( a \) determines the angle and direction of the line.
- The y-intercept \( b \) is where the line crosses the y-axis.
When \( a eq 0 \), the function maintains a clear one-to-one relationship between inputs and outputs, making it eligible to have an inverse.
Bijective
A bijective function is a type of function that is both one-to-one and onto. These qualities are essential for a function to have an inverse. Let鈥檚 break it down:
For the linear function \( f(x) = ax + b \) where \( a eq 0 \), this bijectiveness is naturally met. Its structure ensures that each \( x \) has one unique \( f(x) \) and covers all possible real numbers, thus making an inverse possible.
- One-to-One (Injective): Each input maps to a distinct output. There are no two different inputs mapped to the same output.
- Onto (Surjective): Every possible output is covered by the function. The outputs span the entire range that is intended.
For the linear function \( f(x) = ax + b \) where \( a eq 0 \), this bijectiveness is naturally met. Its structure ensures that each \( x \) has one unique \( f(x) \) and covers all possible real numbers, thus making an inverse possible.
Constant Function
A constant function is defined simply as: \[ f(x) = c \] where \( c \) is a constant value. Regardless of the input, the output remains \( c \).
In essence, a constant function lacks the unique mapping required for each input-output pair that an inverse demands.
- This results in a horizontal line on a graph.
- Commonly used to represent situations where there is no change or variation through time or conditions.
In essence, a constant function lacks the unique mapping required for each input-output pair that an inverse demands.
One-to-One Function
Understanding one-to-one functions is key for diving deeper into function inverses. In a one-to-one function, each element of the domain (input) is paired with a unique element of the range (output). No two different inputs should map to the same output.
The linear function \( f(x) = ax + b \) automatically becomes one-to-one if \( a eq 0 \). This is because the slope \( a \) ensures that no two inputs will yield the same output. Consequently, tracing backwards from any output is straightforward, helping solidify the function's invertibility.
- Imagine a line of people where each has a unique number. If one number matches two people, it's not one-to-one.
- Enables clear tracing backward: Given an output, you can trace back to exactly one input.
The linear function \( f(x) = ax + b \) automatically becomes one-to-one if \( a eq 0 \). This is because the slope \( a \) ensures that no two inputs will yield the same output. Consequently, tracing backwards from any output is straightforward, helping solidify the function's invertibility.
