We want to find out a few pieces of information about the given function.

What family does the function belong to?
What is the domain of this function?
What range does the function have?

Let's answer these questions!

Function Family

Look carefully at the function rule.

h (x) = ∣ x - 3 ∣ + 2

We can see that given function is an absolute value. Therefore, it belongs to the family of absolute value functions, with a parent function of $y = ∣ x ∣ .$

Domain

Unless specific restrictions are given, the domain of absolute value functions is usually all real numbers.

Domain : - \infty < x < \infty

Range

The range of absolute value functions depends on the

y -

coordinate of the vertex, so let's find this value first.

h (x) = ∣ x - 3 ∣ + 2

In this type of functions, the vertex is the point in which the

absolute value expression

equals

0 .

∣ x - 3 ∣ = 0 \Leftrightarrow x = 3

Knowing the

x -

coordinate of the vertex, we are able to find its

y -

coordinate.

h (x) = ∣ x - 3 ∣ + 2

Substitute

$x = 3$

h (3) = ∣ 3 - 3 ∣ + 2

SubTerm

Subtract term

h (3) = ∣ 0 ∣ + 2

AbsPos

$∣ 0 ∣ = 0$

h (3) = 2

Since the

y -

coordinate of the vertex equals

2

and the graph opens upwards (because there is no minus sign before absolute value expression), the range of this function is all real numbers greater than or equal to

2 .

Range : y \geq 2

Checking the Answer

To verify your answer, enter the function rule in the calculator by pushing $Y =$ and writing the function rule in the first row. Next, by pushing $GRAPH,$ the calculator will draw the graph of the function.

Looking at the graph, we can confirm that our solution is correct.

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