Problem 50 - Algebra 2

Question

For each function \(f,\) find \(f^{-1},\) the domain and range of \(f\) and \(f^{-1},\) and determine whether \(f^{-1}\) is a function. $$ f(x)=\sqrt{x+2} $$

Step-by-Step Solution

Answer

The inverse function of \(f(x) = \sqrt{x + 2}\) is \(f^{-1}(x) = x^2 - 2\). The domain of \(f\) is \([-2, \infty )\) and its range is \([0, \infty )\). For \(f^{-1}\), the domain is \((- \infty, \infty )\) and the range is \([-2, \infty )\). \(f^{-1}\) is a function.

Step 1: Find the Inverse of the Function

The first step is to find the inverse of the function. To do this, start by setting \(y = f(x) = \sqrt{x + 2}\). This can be rewritten as \(x = y^2 - 2\). Then, swap \(x\) and \(y\) to get the inverse function \(f^{-1}(x) = x^2 - 2\).

Step 2: Determine the Domain and Range of the Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For \(f(x) = \sqrt{x + 2}\), the value inside the square root can't be negative as we can't take square roots of negative numbers in real numbers. Therefore, the domain of \(f(x)\) is \([ -2, \infty )\). As for the range of the function, which is the set of all output values (y-values), every real number square root is non-negative so the range of \(f(x)\) is \([0, \infty )\).

Step 3: Determine the Domain and Range of the Inverse Function

For \(f^{-1}(x) = x^2 - 2\), the function is defined for all real numbers, since we can square any real number and subtract 2. Therefore, the domain of \(f^{-1}(x)\) is \(-\infty, \infty )\). The range, however, is \([-2, \infty)\) because squaring any real number gives a non-negative result and subtracting 2 shifts the result down by 2.

Step 4: Determine Whether the Inverse Is a Function

A relation is a function if and only if every x-value has exactly one y-value associated with it. For \(f^{-1}(x) = x^2 - 2\), for any given \(x\), there is always exactly one \(y = x^2 - 2\), so \(f^{-1}\) is a function.

Key Concepts

Domain and Range

Understanding the domain and range of functions helps us see where a function starts and finishes. The **domain** of a function is a set of all possible values that the input or 'x' can take. For the function given, which is

• \(f(x) = \sqrt{x + 2}\)

you cannot take the square root of a negative number in real-number functions, which restricts the domain. Therefore, the domain of \(f(x)\) is limited to

• \([-2, \infty)\)

since adding 2 to any number ensures the inside of the square root is non-negative.
Meanwhile, the **range** is about all the possible outputs or 'y' values from the function. Since a square root produces non-negative results, the range is

• \([0, \infty)\)

indicating that no negative outcomes for \(f(x)\) are possible.
Analyzing the inverse function, \(f^{-1}(x) = x^2 - 2\),the domain is all real numbers because squaring any number is always defined. However, after squaring and subtracting 2, the range becomes

• \([-2, \infty)\)

understanding these facets is critical in mastering functions.

Square Root Function

The square root function is quite common in mathematics and has distinct characteristics. A square root function is one where the variable under the radical sign is the primary expression, such as

• \( f(x) = \sqrt{x+2} \).

Square root functions are always defined for **non-negative input values** under the root, leading to unique domain and range behaviors.
A key feature of square root functions is that they naturally include an offset to ensure only non-negative values inside the root. In this case, \(+2\) is the offset that makes \(x + 2\) non-negative for its domain of

• \([-2, \infty)\).

Practically, this means:

• For any input \(x\) within the domain, the function generates non-negative outputs, reflecting its range:
• \([0,\infty)\).

So, never expect negative function results from under the square root!

Function Inverse

Finding the inverse of a function can unlock deeper insights into mathematical relationships. To find an inverse, switch the roles of the dependent and independent variables and solve for the new dependent variable.
Take the function

• \( f(x) = \sqrt{x + 2} \)

Setting

• \( y = \sqrt{x + 2} \)

leads to rewriting this to

• \( x = y^2 - 2 \),

the inverse function is therefore:

• \( f^{-1}(x) = x^2 - 2 \).

The discovery shows how 'x' and 'y' swap places, altering the view of input and output relationships.
It's vital to verify that the range of the original function becomes the domain of its inverse, and the process ensures that the inverse truly reflects the original function's behavior backward.
Each real-valued 'x' in the inverse produces exactly one 'y', confirming the inverse remains a function.

Real-Number Functions

Real-number functions work within the realm of **real numbers** where typical arithmetic operations occur. Real-number functions can accept any real number as input

• provided they do not violate conditions like those arising from square roots or divisions by zero.

The given function \( f(x) = \sqrt{x+2} \) operates within the real number system and adheres to the constraint that we can't have negatives inside the root.
The inverse

• \( f^{-1}(x) = x^2 - 2 \)

is also a typical real-number function, allowing for the squaring of & subtracting from any real 'x'.
Remember:

• Real-number functions depict vital use cases in mathematics due to their flexibility and relevance across different real-world applications,
• and understanding them can help illuminate diverse mathematical concepts like equations of lines, circles, and parabolas.

Studying their properties builds a robust foundation in math and science.