Problem 83 - College Algebra with Modeling and Visualization

Question

Find a formula for \(f^{-1}(x) .\) Identify the domain and range of \(f^{-1}\). Verify that \(f\) and \(f^{-1}\) are inverses. $$ f(x)=x^{2}, x \geq 0 $$

Step-by-Step Solution

Answer

\( f^{-1}(x) = \sqrt{x} \); domain and range are \([0, \infty)\). Verified by substitution.

Step 1: Understand the Inverse Function Concept

To find the inverse of a function, we need to swap its input with the output. For a function \( f(x) \), if \( f(y) = x \), the inverse, denoted \( f^{-1}(x) \), satisfies \( f^{-1}(x) = y \). Here, \( x = y^2 \) operations should be reversed to find \( y \).

Step 2: Exchange Variables

Given \( f(x) = x^2 \) and \( x \geq 0 \), replace \( f(x) \) with \( y \) to represent the output: \( y = x^2 \). Now, swap \( x \) and \( y \) to find the inverse: \( x = y^2 \).

Step 3: Solve for the Inverse Function

To solve for \( y \) in terms of \( x \) from \( x = y^2 \), we take the square root of both sides. Since \( x \geq 0 \) for \( f \), \( y \) also needs to be non-negative: \( y = \sqrt{x} \). Thus, \( f^{-1}(x) = \sqrt{x} \).

Step 4: Identify the Domain and Range of \( f^{-1} \)

The domain of \( f^{-1} \) consists of all possible inputs (outputs of \( f \)), which is \([0, \infty)\) because \( f(x) \) outputs non-negative numbers. The range of \( f^{-1} \) is also \([0, \infty)\) since the square root function outputs non-negative values.

Step 5: Verify the Inverse Relationship

To verify \( f \) and \( f^{-1} \) are inverses, show that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \) for \( x \geq 0 \). Substituting \( x \) into \( f \), we get: \( f(f^{-1}(x)) = f(\sqrt{x}) = (\sqrt{x})^2 = x \). For \( f^{-1} \), \( f^{-1}(f(x)) = f^{-1}(x^2) = \sqrt{x^2} = x \). Both yield \( x \), confirming the inverse relationship.

Key Concepts

Domain and Range

The domain and range of a function are fundamental aspects to understand its behavior fully.
The **domain** of a function refers to all the input values for which the function is defined.
Meanwhile, the **range** is all the output values the function can produce. When dealing with inverse functions, the roles of domain and range between a function and its inverse are interchanged.
The function in question, \( f(x) = x^2 \), with \( x \geq 0 \), is defined for non-negative inputs.
Thus, its domain is \([0, \infty)\) and its outputs or range are also non-negative, \([0, \infty)\).
For the inverse function \( f^{-1}(x) = \sqrt{x} \), the domain must be the same as the range of the original function.
Hence, \( f^{-1} \) is defined for \( x \geq 0 \) as well.
Corresponding to its domain, the range of \( f^{-1} \) also spans non-negative numbers, \([0, \infty)\).
This interchange highlights how the input-output elasticity changes once you find an inverse.

Function Verification

To establish if a function and its proposed inverse truly are opposites, we use function verification.
This process involves proving two equations: \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).
They must hold true across the domain of the functions involved.

• First, verify \( f(f^{-1}(x)) = x \).\( f^{-1}(x) = \sqrt{x} \) implies \( f(f^{-1}(x)) = (\sqrt{x})^2 \).
  This results in \( x \), satisfying the first condition.


• Next, check \( f^{-1}(f(x)) = x \). Substituting \( f(x) = x^2 \) gives \( f^{-1}(x^2) = \sqrt{x^2} \).
  This simplifies to \( x \) since we are considering \( x \geq 0 \), meeting the second requirement.

By achieving \( x \) in both scenarios, we ensure that \( f \) and \( f^{-1} \) are indeed inverses.
It guarantees the operations of squaring and taking the square root are inverse processes within the given domain.

Square Root

Taking the square root is a common mathematical operation, symbolically denoted by \( \sqrt{\cdot} \).
It involves finding a number that, when multiplied by itself, gives the original number.
Given \( x \), the square root, \( \sqrt{x} \), provides this outcome.
For non-negative numbers, the square root is always non-negative, reinforcing its utility as an inverse function to squaring.
When \( f(x) = x^2 \), its inverse becomes \( f^{-1}(x) = \sqrt{x} \) because it reverses the effect of squaring.
The square root function has several properties that must be acknowledged:

• It is only defined for non-negative numbers in the real number system.


• Its value doubles when you square the result - often termed as idempotence.


• This operation smoothens the transition from squared values back to base values.

Understanding how the square root operates is essential to mastering inverse functions.
It equips you with a practical method to invert exponential growth given specific constraints.