Question

Find (a) The domain. (b) The range. $$ h(z)=5+\sqrt{z-25} $$

Short answer

Question: Determine the domain and range of the function h(z) = 5 + sqrt(z - 25). Answer: The domain of the function is z ≥ 25, and the range of the function is h(z) ≥ 5.

Step-by-step solution

Find the domain

The domain is the set of all values of z that make the function valid. Since taking the square root of a negative number is not allowed, we must ensure that the value inside the square root is non-negative (greater than or equal to zero). To find the domain, we must solve the following inequality: $$ z - 25 \geq 0 $$ To solve this inequality, simply add 25 to both sides: $$ z \geq 25 $$ The domain of the function is all z values greater than or equal to 25.

Calculate the Range

Now that we have found the domain, we can find the range, which is the set of all possible output values of h(z). Since the square root function always returns non-negative values and we're adding 5 to the result, the minimum value of the square root is 0 and the minimum value of h(z) is 5 (when z = 25). As z increases, the square root increases, and the value of h(z) also increases. There is no upper bound to the value of h(z). Thus, the range is: $$ h(z) \geq 5 $$ The range of the function is all h(z) values greater than or equal to 5.

Key concepts

Understanding Function Range

The range of a function is the complete set of possible output values, based on the given domain inputs. For the function \( h(z) = 5 + \sqrt{z-25} \), understanding its range requires knowing what happens when you substitute domain values into the function.
The square root of a number is always non-negative. Therefore, when \( z = 25 \) (the smallest value in the domain), the inside of the square root becomes 0, yielding \( \sqrt{0} = 0 \). Adding 5 gives us the lowest value of \( h(z) \) as 5.
As \( z \) becomes larger than 25, the value inside the square root increases, making \( \sqrt{z-25} \) grow larger. This, in turn, causes \( h(z) \) to grow without limit. Consequently, the range of \( h(z) \) starts at 5 and extends to infinity, formally written as \( h(z) \geq 5 \).
This understanding of ranges is crucial when dealing with any function, helping to predict all potential outputs.

Solving Inequalities

Inequality solving is a fundamental skill when dealing with expressions within functions, especially when determining domains. In our specific problem, we dealt with the inequality \( z - 25 \geq 0 \).
This stems from our requirement that what we're taking the square root of must be non-negative.
Solving such inequalities straightforwardly follows typical algebraic rules, like adding or subtracting constants.
So, in our case, adding 25 to both sides of \( z - 25 \geq 0 \) is the solution:

• Initial inequality: \( z - 25 \geq 0 \)
• Add 25: \( z \geq 25 \)

This shows that \( z \) must be 25 or higher for the function to be defined. This kind of problem often appears when solving domain-related issues, emphasizing the importance of properly handling inequalities.

Exploring the Square Root Function

The square root function, symbolized as \( \sqrt{x} \), is a function that assigns to each non-negative number \( x \) the non-negative number \( y \) such that \( y^2 = x \). Within our function \( h(z) = 5 + \sqrt{z-25} \), the square root function has intrinsic constraints.
Because you cannot take a square root of a negative number without involving complex numbers, the input for a square root must be zero or greater. This restriction heavily influences the domain of functions involving square roots.
The basic behavior of the square root function is:

• Always non-negative: Results are zero or positive.
• Increasing: As the input grows, the output grows slower due to diminishing returns.

The range of the square root function typically involves only non-negative numbers, providing a base for other functions to build upon, as seen in our problem where starting at zero in the square root shifts \( h(z) \) to start at a minimum value of 5.