Question

Given the following absolute value function sketch the graph of the function and find the domain and range.

ƒ(x) = |x + 3| - 1

pls show how did u solve it

Answer 1

• Vertex = (-3, -1).
• y-intercept = (0, 2).
• x-intercepts = (-2, 0) and (-4, 0).
• Domain = (-∞, ∞).
• Range = [-1, ∞).

Explanation:

Given absolute value function:

`f(x)=|x+3|-1`

The parent function of the given function is:

`f(x)=|x|`

Graph of the parent absolute function:

• Line |y| = -x where x ≤ 0
• Line |y| = x where x ≥ 0
• Vertex at (0, 0)

Translations

`f(x+a) \implies f(x) \: \textsf{translated $a$ units left}.`

`f(x-a) \implies f(x) \: \textsf{translated $a$ units right}.`

`f(x)+a \implies f(x) \: \textsf{translated $a$ units up}.`

`f(x)-a \implies f(x) \: \textsf{translated $a$ units down}.`

Therefore, the given function is the parent function translated 3 units left and 1 unit down.

If the vertex of the parent function is (0, 0) then the vertex of the given function is:

⇒ Vertex = (0 - 3, 0 - 1) = (-3, -1)

To find the y-intercept, substitute x = 0 into the given function:

`\implies \textsf{$y$-intercept}=|0+3|-1=2`

To find the x-intercepts, set the function to zero and solve for x:

`\implies |x+3|-1=0`

`\implies |x+3|=1`

Therefore:

`\implies x+3=1 \implies x=-2`

`\implies x+3=-1 \implies x=-4`

Therefore, the x-intercepts are (-2, 0) and (-4, 0).

To sketch the graph:

• Plot the found vertex, y-intercept and x-intercepts.
• Draw a straight line from the vertex through (-2, 0) and the y-intercept.
• Draw a straight line from the vertex through (-4, 0).
• Ensure the graph is symmetrical about x = -3.

Note: When sketching a graph, be sure to label all points where the line crosses the axes.

The domain of a function is the set of all possible input values (x-values).

The domain of the given function is unrestricted and therefore (-∞, ∞).

The range of a function is the set of all possible output values (y-values).

The minimum of the function is the y-value of the vertex: y = -1.

Therefore, the range of the given function is: [-1, ∞).

Answer 2

Given

• Absolute value function f(x) = |x + 3| - 1

To find

• Sketch the graph,
• Find its domain,
• Find its range.

Solution

In order to sketch the graph we need to find the vertex and two more points to connect with the vertex.

To do so set the inside of absolute value to zero:

• x + 3 = 0
• x = - 3

The y-coordinate of same is:

• f(-3) = 0 - 1 = - 1.

So the vertex is (- 3, - 1).

Since the coefficient of the absolute value is positive, the graph opens up, and the vertex is below the x-axis as we found above.

Find the x-intercepts by setting the function equal to zero:

• |x + 3| - 1 = 0
• x + 3 - 1 = 0 or - x - 3 - 1 = 0
• x + 2 = 0 or - x - 4 = 0
• x = - 2 or x = - 4

We have two x-intercepts (-4, 0) and (-2, 0).

Now plot all three points and connect the vertex with both x-intercepts.

Now, from the graph we see there is no domain restrictions but the range is restricted to y-coordinate of the vertex.

It can be shown as:

• Domain: x ∈ ( - ∞, + ∞),
• Range: y ∈ [ - 1, + ∞)