Problem 17 - Algebra and Trigonometry

Question

Assume \(f\) is a one-to-one function. (a) If \(f(2)=7,\) find \(f^{-1}(7)\) (b) If \(f^{-1}(3)=-1,\) find \(f(-1)\)

Step-by-Step Solution

Answer

(a) \( f^{-1}(7) = 2 \); (b) \( f(-1) = 3 \).

Step 1: Understanding One-to-One Functions

A one-to-one function ensures that each output value of the function corresponds to exactly one input value. This property implies that if you know the output, you can reliably find the single input value.

Step 2: Finding the Inverse Function Value for Part (a)

Given that \( f(2) = 7 \), to find \( f^{-1}(7) \), we recognize that because \( f \) is a one-to-one function, the inverse will map the output back to its original input. Hence, \( f^{-1}(7) = 2 \).

Step 3: Finding Function Value for Part (b) Using the Inverse

We are given \( f^{-1}(3) = -1 \). This means that when the input to the inverse function is 3, the output is -1, which, by the definition of inverse functions, implies that \( f(-1) = 3 \).

Key Concepts

One-to-One Function

A one-to-one function, also known as an injective function, is a special type of function that pairs each output value with exactly one input value. Think of it as a unique matchmaking system where each output has only one soulmate among the inputs.
This means no two different inputs ever point to the same output.
In mathematical terms, for a function to be one-to-one, if \( f(a) = f(b) \) then it must also be true that \( a = b \).
This guarantee allows us to work backwards to find the input if the output is known.

• Each input results in a unique output.
• If outputs repeat for different inputs, the function is not one-to-one.

Understanding this concept helps when working with inverse functions, as the process of finding the original input from a known output depends on this unique relationship.

Function Inverse

Every function that is one-to-one has an inverse function, denoted as \( f^{-1} \). This inverse swaps the roles of inputs and outputs.
For instance, if a function \( f \) takes an input \( x \) and produces an output \( y \), then the inverse function \( f^{-1} \) should take \( y \) as input to give back \( x \) as output.Finding an inverse requires reversing the original function’s process.
Practically, this means for a pair \((a, b)\) in the function \( f \), you have \((b, a)\) in \( f^{-1} \).

• Use "undo" operations: If \( f(x) = y \), then \( f^{-1}(y) = x \).
• The graph of \( f^{-1} \) is a mirror of the graph of \( f \) about the line \( y = x \).

Understanding inverse functions is crucial for solving problems where the starting point and ending point are reversed, requiring one to trace backwards through the function.

Algebraic Functions

Algebraic functions comprise polynomials and other expressions formed using basic operations including addition, subtraction, multiplication, division, and taking roots.
Examples include quadratic, cubic, or more complex expressions like \( f(x) = x^2 + 3x + 2 \).Such functions often involve solving equations to find values for \( x \) that result in a specific \( y \).
They are versatile and allow modeling a vast array of real-world phenomena in mathematics.Key Properties:

• Smoothness and continuity mean these functions typically don't have abrupt changes.
• Can take various forms, including linear and nonlinear ones.

Understanding these functions is essential, as they offer a foundational framework for many advanced mathematical concepts, including calculus and differential equations.