Lesson 1: The Square Root Function

Property

A square root function is a function that is defined by a square root expression.
To evaluate a square root function, we find the value of $f(x)$ for a given value of $x$ by substituting the value and simplifying the square root.

Examples

• For the function $f(x) = \sqrt{2x - 1}$, to find $f(5)$, substitute 5 for $x$: $f(5) = \sqrt{2(5) - 1} = \sqrt{9} = 3$.
• For the function $g(x) = \sqrt{x + 7}$, to find $g(2)$, substitute 2 for $x$: $g(2) = \sqrt{2 + 7} = \sqrt{9} = 3$.
• For the function $h(x) = \sqrt{3x + 6}$, to find $h(-3)$, substitute -3 for $x$: $h(-3) = \sqrt{3(-3) + 6} = \sqrt{-3} = \sqrt{3}$.

Explanation

A square root function is a function that contains a square root symbol with the variable inside the radical. To evaluate it, you substitute the given number for the variable $x$ and then simplify the expression under the square root symbol.

Property

Examples

• The function $f(x) = \sqrt{x}$ is undefined for negative inputs, so its domain starts at 0. For example, $f(25) = 5$ and $f(9) = 3$.
• Since we can only take the square root of non-negative numbers, the domain is $[0, \infty)$. For instance, $f(0) = 0$, $f(4) = 2$, and $f(16) = 4$.
• The range is $[0, \infty)$ because the square root function can never produce a negative output value. The smallest possible output is $f(0) = 0$.

Explanation

The square root function $f(x) = \sqrt{x}$ starts at the point $(0,0)$ and accepts only non-negative inputs. This restriction exists because we cannot take the square root of negative numbers in the real number system. The function produces only non-negative outputs, creating a curved graph that increases gradually as $x$ increases.

Property

The graph of $f(x) = \sqrt{x} + k$ shifts the graph of $f(x) = \sqrt{x}$ vertically $k$ units.

• If $k > 0$, shift the square root function vertically up $k$ units.
• If $k < 0$, shift the square root function vertically down $|k|$ units.

Examples

Property

The graph of $f(x) = \sqrt{x - h}$ shifts the graph of $f(x) = \sqrt{x}$ horizontally $h$ units.

• If $h > 0$, shift the graph horizontally right $h$ units.
• If $h < 0$, shift the graph horizontally left $|h|$ units.

Examples