Question

In Exercises 49 and \(50,\) (a) draw the graph of the function. Then find its (b) domain and (c) range. $$f(x)=-|3-x|+2$$

Short answer

The domain of the function \(f(x)=-|3-x|+2\) is \(-\infty < x < \infty\). The range of the function is \(-\infty < y ≤ 2\).

Step-by-step solution

Drawing the graph

To graph the function \(f(x)=-|3-x|+2\), start by graphing the base function, which is the absolute value function, \(|x|\). This will be in the shape of a V with a minimum point at (0,0). However, the absolute value function in this exercise is \(|3-x|\), which shifts the graph to the right by 3 units. Taking the negative of this function, \(-|3-x|\), reflects the graph over the x-axis. Lastly, adding 2, \(-|3-x|+2\), shifts the graph up by 2 units.

Finding the Domain

The domain of a function is the set of all possible x-values for which the function is defined. As there is no denominator or square root in our function, there are no x-values that will make the function undefined. Therefore, the domain of this function is all real numbers, written as \(-\infty < x < \infty\).

Finding the Range

The range of a function is the set of all possible y-values for the function. Looking at the graph from Step 1, you can see that the highest point on the graph is (3,2). This is where the vertex of the function is. Therefore the range of the function is \(-\infty < y ≤ 2\).

Key concepts

Domain of a Function

Understanding the domain of a function is key to unlocking the behavior and limitations of mathematical equations. The domain represents all the possible input values, or 'x' values, that you can substitute into a function without creating undefined conditions. For instance, when dealing with fractions, we have to avoid zero denominators, and for square roots, the radicand must be non-negative.

Looking at absolute value functions, such as our example function \( f(x)=-|3-x|+2 \), we can see there are no such restrictions. Therefore, the domain of this function encompasses all real numbers. Mathematically, we express this as \( -\infty < x < \infty \). It’s important to realize that no matter what value 'x' assumes, the function will always yield a valid result. This comprehensive domain is characteristic of absolute value functions and is crucial when graphing such functions or considering potential inputs for real-world applications.

Range of a Function

While the domain of a function focuses on input values, the range is all about output. Specifically, the range of a function consists of all possible 'y' values (outputs) that result from plugging the domain (inputs) into the function. Determining the range of a function, such as \( f(x)=-|3-x|+2 \), requires a close look at its graph, particularly the highest and lowest points the output can achieve.

For our absolute value function, the graph opens downward due to the negative sign in front, with its vertex acting as the highest point at \(y=2\). This signifies that the function's value will never exceed 2. Hence, the range of our function is \( -\infty < y \leq 2 \). No y-value in this function will be greater than 2, which guides us when graphing the function and interpreting its real-world implications, as it gives us a ceiling for our expected outcomes.

Transformations of Functions

Transformations are an invaluable tool in mathematics, allowing us to take basic functions and manipulate them to create entirely new graphs. Let's examine the transformations that shape our example function \( f(x)=-|3-x|+2 \).

The base function, \( |x| \), which forms a V-shape, undergoes multiple transformations: a horizontal shift, reflection, and vertical shift.

• A horizontal shift occurs due to the \( |3-x| \) component, moving the graph 3 units to the right.
• The reflection happens with the negative sign before the absolute value, flipping the graph over the x-axis, making it open downwards instead of upwards.
• Finally, the vertical shift comes from the '+2', lifting the graph 2 units higher.

Understanding these transformations is crucial for graphing more complex functions and can reveal more about the function's behavior in relation to its basic form.