Even Roots

This course is the study of the real numbers. As a result we don’t take even roots of negative numbers.

Even roots

$$Even(x) = \sqrt [2n]{x} = x^{\tfrac {1}{2n}}$$

look a lot like the square root

$$Even(x) = \sqrt {x} = x^{\tfrac {1}{2}}$$

Their domains do not include negative numbers: $[0, \infty )$. Their domains are the nonnegative numbers. These functions increase very slowly over their domain, becoming unbounded.

The graph of $y = SR(x) = \sqrt {x}$

The square root function begins where the inside of the radical equals $0$. It then moves in the direction that keeps the inside positive.

Even Root

The graph of $y = f(v) = \sqrt {v+3}$

Here the inside $v+3$ equals $0$ when $v=\answer {-3}$. That is the start of the domain. Then, the inside is positive $v+3>0$, when $v>-3$, which means the graph moves up to the right.

Even Root

The graph of $y = T(k) = \sqrt {-k+5}$

Here the inside $-k+5=0$ equals $0$ when $k=\answer {5}$. That is the start of the domain. Then the inside is positive $-k+5>0$, when $k$ $<$ $>$ $5$, which means the graph moves up to the left. The domain is $(-\infty ,5]$.

Odd Roots

This course is the study of the real numbers. As a result we don’t take even roots of negative numbers, however we do have odd roots of negative numbers

Odd roots

$$Odd(x) = \sqrt [2n+1]{x} = x^{\tfrac {1}{2n+1}}$$

look a lot like the cube root

$$Odd(x) = \sqrt [3]{x} = x^{\tfrac {1}{3}}$$

Their domains include all real numbers: $(-\infty , \infty )$. They increase very slowly over this domain, becoming unbounded.

The graph of $y = CubeRoot(x) = \sqrt [3]{x}$

The cube root function has a vertical tangent line where the inside of the radical equals $0$.

Odd Root

The graph of $y = f(v) = \sqrt [3]{v+3}$

Here the inside $v+3$ equals $0$ when $v=\answer {-3}$. The graph has a vertical tangent line there. Otherwise, the cube root function is increasing everywhere, but increasing slower and slower and slower.