Answer:
To find the range of the function y=x3−3y=3x
−3, we need to analyze how the function behaves over the given domain −8<x<8−8<x<8.
Explanation:
First, let's look at the behavior of x33x
. The cube root function is defined for all real numbers, so there are no restrictions on the domain of x33x
. However, the cube root function has a continuous graph and passes through (0,0)(0,0), (−1,−1)(−1,−1), and (1,1)(1,1). As xx becomes more negative or more positive, x33x
will approach negative infinity or positive infinity, respectively.
Now, we subtract 3 from x33x
to get the function y=x3−3y=3x
−3. This will shift the graph of x33x
downward by 3 units.
Since the cube root function approaches positive and negative infinity for extreme values of xx, subtracting 3 from it will also result in a range that extends to positive and negative infinity.
Thus, the correct range of the graph is:
{y ∣ −∞<y<∞}{y∣−∞<y<∞}
However, none of the given options match this answer exactly. The closest option is {y ∣ −5<y<5}{y∣−5<y<5}, which includes all real numbers between -5 and 5, but it doesn't account for values below -5 or above 5.
Therefore, the best answer among the options provided would be:
{y ∣ −5<y<5}{y∣−5<y<5}