Question

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

Answer 1

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`.