Radical function
The square root of a negative number does not exist within the real numbers. This has a direct consequence for graphs: the radical function cannot be drawn just anywhere, only where the expression inside the root is not negative. Finding out where the curve starts is the first question to ask.
The domain: what is inside cannot be negative
Determine the domain of definition of the function $f(x) = \sqrt{4x - 12}$.
Determine the domain of definition of the function $f(x) = \sqrt{4x - 12}$.
Write the algebraic expression of the function resulting from translating the graph of the function $f(x) = \sqrt{x}$ three units to the left and five units down.
Write the algebraic expression of the function resulting from translating the graph of the function $f(x) = \sqrt{x}$ three units to the left and five units down.
Calculating images
To find $f(a)$, substitute $x=a$ into the radicand and calculate the root. If the radicand turns out negative, $a$ does not belong to the domain and the image does not exist.
Given the function $h(x) = \sqrt{5 - 2x}$, calculate the images corresponding to the values $x = -2$, $x = \frac{1}{2}$ and $x = -10$.
Given the function $h(x) = \sqrt{5 - 2x}$, calculate the images corresponding to the values $x = -2$, $x = \frac{1}{2}$ and $x = -10$.
Translations of $y=\sqrt{x}$
Look at the following graphs and match each of them (A, B, C and D) with its corresponding algebraic expression from the following options: $y = \sqrt{x+2}$,…
Look at the following graphs and match each of them (A, B, C and D) with its corresponding algebraic expression from the following options: $y = \sqrt{x+2}$, $y = \sqrt{x} + 2$, $y = \sqrt{x-2}$, and $y = \sqrt{x} - 2$.
Determine the domain of definition of the function $f(x) = \sqrt{-x^2 + 2x + 8}$.
Determine the domain of definition of the function $f(x) = \sqrt{-x^2 + 2x + 8}$.
Intercepts with the axes
With the $Y$-axis: calculate $f(0)$, if $0$ is in the domain. With the $X$-axis: solve $f(x)=0$, squaring both sides of the equation to eliminate the root.
Find the axis intercepts of the function $g(x) = \sqrt{3x + 9} - 3$.
Find the axis intercepts of the function $g(x) = \sqrt{3x + 9} - 3$.
The period $T$ (in seconds) of a simple pendulum is given by the function $T(l) = 2\pi\sqrt{\frac{l}{9,8}}$, where $l$ is the length of the pendulum in metres. Calculate…
The period $T$ (in seconds) of a simple pendulum is given by the function $T(l) = 2\pi\sqrt{\frac{l}{9,8}}$, where $l$ is the length of the pendulum in metres. Calculate the period of a pendulum of length $2,5\text{ m}$ and find out what length a pendulum must have so that its period is exactly $2\text{ s}$.
The following graph corresponds to a function of the type $f(x) = a\sqrt{x-b} + c$. Look at the key points marked on the curve and determine the exact value of the…
The following graph corresponds to a function of the type $f(x) = a\sqrt{x-b} + c$. Look at the key points marked on the curve and determine the exact value of the parameters $a$, $b$ and $c$.
Back to the start: To find out exactly where the curve begins, set the radicand equal to zero ($g(x)=0$); the graph will start at that value of $x$ and will only exist on the side where the radicand is positive or zero.