Problem 25 - Algebra 2

Question

For each function \(f,\) find \(f^{-1}\) and the domain and range of \(f\) and \(f^{-1} .\) Determine whether \(f^{-1}\) is a function. $$ f(x)=\sqrt{x+7} $$

Step-by-Step Solution

Answer

The inverse of the function \(f(x)=\sqrt{x+7}\) is \(f^{-1}(x) = x^2 - 7\), and it is a function. The domain and range of \(f(x)\) are \([-7, \infty)\) and \([0, \infty)\) respectively. The domain and range of \(f^{-1}(x)\) are \(-\infty, \infty)\) and \([-7, \infty)\) respectively.

Step 1: Finding the Inverse

To find the inverse \(f^{-1}(x)\) , replace \(f(x)\) with \(y\) , then swap \(x)\ and \(y\) and finally solve for \(y\). So, \(y = \sqrt{x+7}\) becomes \(x = \sqrt{y+7}\). Now solving for \(y\), we square both sides: \(x^2 = y+7\) , which simplifies to \(y = x^2 - 7\). So, \(f^{-1}(x) = x^2 - 7\).

Step 2: Checking if the Inverse is a Function

A real function must have only one output value (y) for each of its input value (x). Looking at the inverse \(f^{-1}(x) = x^2 - 7\), we can see that for each input of x, there is only one corresponding output value of y. Hence, the inverse is indeed a function.

Step 3: Finding the Domain and Range of the Original Function and its Inverse

For \(f(x) = \sqrt{x+7}\), the domain will be \([-7, \infty)\) because x+7 must be greater than or equal to 0. The range will be \([0, \infty)\) as square root is always positive or 0. For the inverse function \(f^{-1}(x) = x^2 - 7\), the domain will be \(-\infty, \infty)\) as there are no restrictions on x, and the range will be \([-7, \infty)\) as x squared minus 7 will always be greater than or equal to -7.

Key Concepts

Domain and Range

Understanding domain and range is crucial to grasping functions and their inverses. The domain of a function is the set of all possible input values (x-values) that the function can accept. For the function

• \( f(x) = \sqrt{x+7} \), the domain is determined by ensuring that the expression under the square root is non-negative, because we cannot take the square root of a negative number in the real number system.
• Thus, the domain is \([-7, \infty)\), meaning x can be any value greater than or equal to -7.

The range, on the other hand, is the set of all possible output values (y-values) you get by plugging the domain values into the function.

• For \( f(x) = \sqrt{x+7} \), since the square root of any non-negative number is also non-negative, the range is \([0, \infty)\).
• For the inverse function \( f^{-1}(x) = x^2 - 7 \), the inverse operation opens up all real x-values, giving it a domain of \((-\infty, \infty)\).
• The range will be \([-7, \infty)\) because squaring any real x, and then subtracting 7 from it, results in values starting at -7 and increasing indefinitely.

Understanding these concepts helps in identifying the limitations and possibilities within which the functions can operate.

Square Root Function

The square root function is a special kind of function that returns the square root of the input number. It's often expressed as \(f(x) = \sqrt{x} \), which transforms quickly when adjusted to \( f(x) = \sqrt{x+7} \).

• This kind of function only provides outputs that are zero or positive, because a square root cannot yield a negative number in the realm of real numbers.
• When graphed, the square root function starts from a point, then rises gradually to the right, looking like half of a parabola rotated sideways.
• It's essential to note that, the domain of the function \( f(x) = \sqrt{x+7} \) is \([ -7, \infty)\).
• This is because the expression inside the square root, \(x+7\), must remain positive, ensuring the function is defined for real-number outputs. The range is always non-negative, \([0, \infty)\).

Square root functions, due to their nature, restrict the inverse function possibilities by having a limited original domain.

Function Verification

Verification of a function and its inverse ensures that the original function and the inverse are valid and correctly formulated.

• The inverse of a function is ideally the reflection of each point of the original function across the line \(y=x\).
• In our exercise, we verified the inverse function \( f^{-1}(x) = x^2 - 7 \).

Verification involves checking whether each input value leads to a single output, confirming that

• each element in the domain of the original has a unique partner in the range.
• Similarly, test if \( f^{-1}(x) \), which represents solving \(x = \sqrt{y+7} \), yields a single valid y for every x.

For \( f^{-1}(x) = x^2 - 7 \), this holds true: every real number x results in a unique output.

• This tells us that \( f^{-1}(x) \) is indeed a function.

Understanding and verifying these fundamentals ensures proper mathematical operations, helping to build a strong foundation in function analysis.