How do you make a table and graph for #h(x) = x^3-4#? 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)# (the cube root of 4) or about #1.59#
Y-intercept: #-4#

Explanation:

1. Table
Here's a table for #h(x)=x^3-4#, but only for #-5<=x<=5#:
|#x# | #h(x)#|
|#-5# | #-129#|
|#-4#|#-68#|
|#-3#|#-31#|
|#-2#|#-12#|
|#-1#|#-5#|
|#0#|#-4#|
|#1#|#-3#|
|#2#|#4#|
|#3#|#23#|
|#4#|#60#|
|#5#|#121#|

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

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

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

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