How do you make a table and graph for h(x) = x^3-4h(x)=x3−4? What is the domain, range, and intercepts?

1 Answer
The Legend
Feb 22, 2018

Table: See below.
Graph: See below.
Domain: All real numbers.
Range: All real numbers.
X-intercept: root(3,4)3,4√ (the cube root of 4) or about 1.591.59
Y-intercept: -4−4

Explanation:

1. Table
Here's a table for h(x)=x^3-4h(x)=x3−4, but only for -5<=x<=5−5≤x≤5:
|xx | h(x)h(x)|
|-5−5 | -129−129|
|-4−4|-68−68|
|-3−3|-31−31|
|-2−2|-12−12|
|-1−1|-5−5|
|00|-4−4|
|11|-3−3|
|22|44|
|33|2323|
|44|6060|
|55|121121|

2. Graph
Here's the graph:
graph{x^3-4 [-16.02, 16.02, -11.43, 4.59]}

3. Domain
The domain is all the values of xx that don't make the equation undefined. This is usually all real numbers, but if you have a fraction with xx in the denominator or a radical with xx inside it, it is not. Neither of these are true, so the domain is all real numbers.

4. Range
The range is all possible yy-values. A cube root can be negative (such as -2^3=-8−23=−8), so the range is also all real numbers.

5. Intercepts
The yy-intercept is -4−4, as the table above shows. The xx-intercept is more complicated. Let's set h(x)h(x) (=x^3-4=x3−4) to zero and solve:
x^3-4=0x3−4=0
Add 44 to both sides:
x^3=4x3=4
Take the cube root of both sides:
x=root(3,4)x=3,4√
x~~1.59x≈1.59
Our xx-intercept is root(3,4)3,4√ or about 1.591.59.

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