Problem 43 - Algebra A Combined Function

Question

Without graphing, find the domain of each function. $$ h(x)=5|x-20|+1 $$

Step-by-Step Solution

Answer

The domain of the function is all real numbers.

Step 1: Understand the function structure

The function given is \( h(x) = 5|x-20| + 1 \). This function consists of an absolute value expression \(|x-20|\), a multiplication by 5, and then an addition of 1.

Step 2: Find the domain of the absolute value expression

The absolute value function \(|x-20|\) can accept any real number as an input. Absolute values are defined for all real numbers because for any real number, you can take its absolute value.

Step 3: Determine the domain of any transformations

The operations of multiplying by 5 and adding 1 do not impose any restrictions on the set of possible inputs. Therefore, these transformations do not affect the domain of the absolute value function.

Step 4: Compile the domain

Since there are no restrictions from the absolute value function or the other components of the function, the domain of \( h(x) = 5|x-20| + 1 \) is all real numbers.

Key Concepts

Absolute Value Function

An absolute value function, represented as \(|x-a|\), measures the distance of a number \(x\) from a particular point \(a\) on a number line. This means, regardless of whether \(x\) is larger or smaller than \(a\), the output is always positive or zero. As a result, absolute value functions can handle any \(x\) from the set of real numbers. Absolute values are particularly useful because they allow functions to express differences without concerning themselves about direction - whether \(x\) is more than or less than \(a\). For instance, \(|3-5| = 2\) and \(|5-3| = 2\). In the exercise function, \(h(x) = 5|x-20| + 1\), the absolute value part \(|x-20|\) ensures the core expression can work with any real number.

Real Numbers

Real numbers encompass nearly all the numbers you generally use in everyday life, including both rational and irrational numbers. They comprise integers, fractions, and decimals. As such, they also include numbers like \(-3,\) \(0,\) \(2.75,\) \(\frac{5}{3}\), and \(\sqrt{2}\). In terms of function domains, the real number set is usually the default unless limitations arise. These can come from expressions such as division by zero or taking square roots of negative numbers in real scenarios. The exercise aims to highlight that there are no such restrictions applied to the absolute value function or its transformations. Hence, it implies that for \(h(x) = 5|x-20| + 1\), the domain is unrestricted, meaning it covers all real numbers.

Function Transformations

Function transformations involve adjusting the basic parameters of a function to change its shape, position, and general behavior. They can include translating, scaling, reflecting, and even rotating the graph of a function. In the context of the exercise function, \(h(x) = 5|x-20| + 1\), you deal with both scaling and translating:

• **Scaling** changes the size of the graph. Here, the absolute value \(|x-20|\) is multiplied by 5, stretching it vertically.
• **Translation** shifts the graph. The "+1" at the end raises the entire graph upward by one unit.

These transformations, however, do not affect the domain of the function. Since they don't introduce factors like division by zero or square roots, the input can still be any real number. Because of this, such transformations are instrumental in shaping the function's output while maintaining a wide input range.