Question

What is the domain and range of the function y = xn if n is an odd whole number? if n is an even whole number?

A. If n is odd, domain is (−∞,∞) and range is [0, ∞); if n is even, domain and range are both (−∞,∞).
B. If n is odd, domain and range are both (−∞, ∞); if n is even, domain is (−∞, ∞) and range is [0, ∞).
C. In both cases, domain is (−∞, ∞) and range is [0, ∞).
D. Domain and range in both cases is (−∞ , ∞ ).

Answer 1

Domain and range are the possible x and y values, a function can take.

For the given functions:

1. If n is odd, domain and range are both (−∞, ∞)
2. If n is even, domain is (−∞, ∞) and range is [0, ∞)

Given that:

$y = x^n$

When n is even.

Assume that $n=2$

The equation becomes:

$y =x^2$

See attachment for the graph of $y =x^2$.

From the graph, we can see that values of x extend to both sides of the x-axis.

So, the domain is: $(-\infty, \infty)$

However, the value of y starts from 0 and the curve opens upward. This means that y only take positive values.

So, the range is: $[0, \infty)$

When n is odd.

Assume that $n =3$

The equation becomes: $y = x^3$

See attachment for the graph of $y = x^3$.

From the graph, we can see that values of x and y extend to both sides of the x-axis and y-axis, respectively.

So, the domain is: $(-\infty, \infty)$

The range is: $(-\infty, \infty)$

Hence, option (B) is correct.

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