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The square root function
Example 1
Where does function \latex{x\mapsto x^2} take the value of \latex{ 4? }
Solution
The answer is: at two places, at \latex{ 2 } and at \latex{ –2 }. (Figure 43)
According to the conventions the positive value, i.e. \latex{ 2 } is denoted as follows: \latex{\sqrt{4}=2} (principal square root of four), and according to this the negative value is denoted as follows: \latex{-\sqrt{4}=-2}.
\latex{y=x^2}
Figure 43
\latex{ 4 }
\latex{ 1 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ -1 }
\latex{ -2 }
\latex{ -3 }
\latex{ y }
\latex{ x }
In general if an \latex{a \gt0} number is given, then there are two places, where function \latex{x^2} takes the value of \latex{a}, out of these \latex{\sqrt{a}} is positive and \latex{-\sqrt{a}} is negative. Function \latex{x^2} takes the value of 0 at only one place, at 0. (Figure 44)
Figure 44
\latex{y=x^2}
\latex{\sqrt{a}}
\latex{-\sqrt{a}}
\latex{a}
\latex{y}
\latex{x}
⯁ ⯁ ⯁
Based on this we accept the following:
DEFINITION: If \latex{a\geq 0}, then \latex{\sqrt{a}} denotes the non-negative number the square of which is a.
Function \latex{f:\R^+_0\rightarrow \R}, \latex{f(x)=\sqrt{x}} is called the square root function. \latex{\R^+_0} denotes the set of non-negative real numbers. So the square root function maps the set of non-negative real numbers to the set of real numbers.
Let us plot the graph of and characterise function \latex{x\mapsto\sqrt{x}}.
We assume that the image of the function \latex{x\mapsto\sqrt{x}} is the half of a parabola the vertex of which is the origin and the axis of which is the \latex{x-}axis. It is not by accident, since we are going to show that the image of the function \latex{x\mapsto\sqrt{x}} is the reflection of the image of the function \latex{x\geq0}, \latex{x\mapsto x^2} about straight line \latex{y=x}. (Figure 45)
\latex{y=\sqrt{x}}
Figure 45
\latex{ 2 }
\latex{ 1 }
\latex{ 1 }
\latex{ 4 }
\latex{ y }
\latex{ x }
Observe that if point \latex{P(a; b)} is reflected about straight line \latex{y=x}, then the reflection image is \latex{P'(b; a)} i.e. the two coordinates are swapped. (Figure 46)
Let us take point \latex{P(a; b)}. It lies on the graph of function \latex{x\mapsto \sqrt{x}} if and only if \latex{\sqrt{a}=b \;(a\geq 0)}, which is fulfilled if and only if \latex{a=b^2}, i.e. point \latex{P'(b; a)} lies on the graph of the function \latex{x^2}. \latex{P} and \latex{P'} are each other's reflection about the straight line \latex{y = x}. (Figure 47)
\latex{y=x}
\latex{P'(b;a)}
\latex{P(a;b)}
\latex{ a }
\latex{ b }
\latex{ b }
\latex{ a }
\latex{ x }
\latex{ y }
Figure 46
The function \latex{x\mapsto\sqrt{x}} is called the inverse function of function \latex{x\geq 0, \; x\mapsto x^2}.
Similarly think it over, that the inverse function of function \latex{x\leq 0}, \latex{x\mapsto x^2} is function \latex{x\geq 0}, \latex{x\mapsto-\sqrt{x}}. The graphs of these are also each other's reflection about the straight line \latex{y = x.} (Figure 48)
The range of function \latex{\sqrt{x}} is the set of non-negative real numbers, and at the edge of its domain, at \latex{ 0 } it has a minimum, its minimum value is \latex{ 0 }.
\latex{P'(b;a)}
\latex{P(a; b)}
\latex{y=x^2}
\latex{y=x}
\latex{y=\sqrt{x}}
\latex{ 1 }
\latex{ 1 }
\latex{ x }
\latex{ y }
Figure 47
\latex{y=x^2}
\latex{y=x}
\latex{y=-\sqrt{x}}
\latex{ 1 }
\latex{ 1 }
\latex{ x }
\latex{ y }
Figure 48
Example 2
Let us plot the graphs of and characterise the following functions, the domain of which is the widest suitable subset of \latex{\R} and the codomain of which is \latex{\R}.
\latex{f(x)=\sqrt{x-3}, \; (x\geq 3)}
\latex{g(x)=\sqrt{x+5}, \; (x\geq -5)}
\latex{h(x)=\sqrt{-x}, \; (x\leq 0)}
\latex{^*k(x)=\sqrt{3-x}, \; (x\leq 3) }
\latex{y=\sqrt{x}}
\latex{y=\sqrt{x-3}}
\latex{ 1 }
\latex{ 1 }
\latex{ 3 }
\latex{ x }
\latex{ y }
Figure 49
Solution
The function \latex{ f } takes the same value at a place \latex{ 3 } greater than function \latex{x\mapsto\sqrt{x}} does. For example it takes \latex{ 1 } at \latex{ 4 } and \latex{ 2 } at \latex{ 7 }.
So its graph is derived by translating the image of the function \latex{x\mapsto\sqrt{x}} by 3 units into the positive direction along the \latex{ x- }axis. (Figure 49)
The range of function f is the set of non-negative real numbers. The function is increasing on interval \latex{[3; +\infty[}, at \latex{ 3 }, at the edge of the domain it has a minimum, its minimum value
is \latex{ 0 }.
So its graph is derived by translating the image of the function \latex{x\mapsto\sqrt{x}} by 3 units into the positive direction along the \latex{ x- }axis. (Figure 49)
The range of function f is the set of non-negative real numbers. The function is increasing on interval \latex{[3; +\infty[}, at \latex{ 3 }, at the edge of the domain it has a minimum, its minimum value
is \latex{ 0 }.
The image of function \latex{ g } is derived similarly: by translating the image of the function \latex{x\mapsto \sqrt{x}} by 5 units into the negative direction along the \latex{ x- }axis. (Figure 50)
The properties of the function can easily be seen in the figure.
The properties of the function can easily be seen in the figure.
\latex{y=\sqrt{x+5}}
\latex{y=\sqrt{x}}
\latex{ 1 }
\latex{ 1 }
\latex{ -5 }
\latex{ x }
\latex{ y }
Figure 50
Function h takes the same values at a place \latex{ –1 } times as function \latex{x\mapsto \sqrt{x}}. For example it takes \latex{ 1 } at \latex{ –1 } and \latex{ 2 } at \latex{ -4 }. So its image is derived by reflecting the image of the function \latex{x\mapsto \sqrt{x}} about the \latex{ y- }axis. (Figure 51)
\latex{y=\sqrt{-x}}
\latex{y=\sqrt{x}}
\latex{ 1 }
\latex{ 1 }
\latex{ x }
\latex{ y }
Figure 51
The formula of function \latex{ k } can be transformed as: \latex{k(x)=\sqrt{3-x}=\sqrt{-(x-3)}}.
It can be seen that the graph of the function \latex{f} should be reflected about the straight line \latex{ x = 3 }. (Figure 52)
\latex{y=\sqrt{3-x}}
\latex{ 1 }
\latex{ 1 }
\latex{ 3 }
\latex{ x }
\latex{ y }
Figure 52
Example 3
Let us plot the graph of and characterise the following \latex{\R\rightarrow\R} function:
\latex{f(x)=\left|x^2-4\right|};
\latex{g(x)=x^2-2\left|x\right|}.
Solution
We have already plotted the image of function \latex{x\mapsto x^2-4}. Function \latex{f} is its absolute value, which means that where \latex{x^2-4\geq0}, there \latex{f} is identical to it, and where \latex{x^2-4\lt0}, there the value of \latex{f} is \latex{ -1 } times it. Thus the corresponding curve part should be reflected about the \latex{ x- }axis. (Figure 53)
Function \latex{ f } is an even function, since \latex{f(–x) = f(x)}, so the curve is symmetric about the \latex{ y- }axis. Thus it is enough to characterise the function for example on interval \latex{[0; +\infty[}, its properties on interval \latex{]–\infty; 0]} result from symmetry.
Function \latex{f} is decreasing on interval \latex{[0; 2]}, it is increasing on \latex{[2; +\infty[}, it has a local maximum at \latex{ 0 } and a minimum at \latex{ 2 }. Its maximum value is 4; its minimum value is \latex{ 0 }. The range of the function is the set of non-negative real numbers.
Function \latex{ g } is also an even function, since \latex{g(–x) = g(x)}, thus it is enough to plot its graph on interval \latex{[0; +\infty[}, and then reflect it about the \latex{ y- }axis.
If \latex{x\geq 0}, then \latex{g(x)=x^2-2x=(x-1)^2-1}, the graph of which can be derived from the graph of \latex{x\mapsto x^2} by translating it by \latex{ 1 } to the right and by \latex{ 1 } downwards.
The monotonicity of the function can easily be followed in Figure 54.
Figure 53
\latex{ -2 }
\latex{ 2 }
\latex{ 4 }
\latex{ x }
\latex{ y }
Figure 54
\latex{ -2 }
\latex{ -1 }
\latex{ 1 }
\latex{ 2 }
\latex{ x }
\latex{ g }
\latex{ y }
Example 4
Let us plot the graph of and characterise the following \latex{\R\rightarrow\R} type function:
\latex{f(x)=\sqrt{\left|x\right|}}.
Figure 55
\latex{y=\sqrt{\left|x\right|}}
\latex{ 1 }
\latex{ 1 }
\latex{ x }
\latex{ y }
Solution
The definition of \latex{ f } can be written piecewise as:
\latex{f(x)=\begin{cases}\,\,\,\,\sqrt{x}, \;\text{if}\; x\geq 0\\ \sqrt{-x}, \; \text{if}\; x\lt0\end{cases}}
According to this the graph of \latex{ f } will be the curve shown in Figure 55.
Exercises
{{exercise_number}}. Plot the graphs of and characterise the following functions.
- \latex{f(x)=\sqrt{-x}, \;(x\leq 0)}
- \latex{g(x)=\sqrt{x}+2, \;(x\geq0)}
- \latex{h(x)=\sqrt{x-2}-2, \; (x\geq 2)}
- \latex{k(x)=\sqrt{x+4}\; (x\geq-4)}
{{exercise_number}}. Plot the graphs of the following \latex{\R\rightarrow\R} type functions, and determine their extreme values and their monotonicity.
- \latex{f(x)=x\left|x\right|}
- \latex{g(x)=\left|x^2-3\left|x\right|+2\right|}
- \latex{h(x)=\sqrt{\left|x-2\right|}}
- \latex{k(x)=x\sqrt{x^2-2x+1}}
{{exercise_number}}. Where are the points with coordinates \latex{(x; y)} in the plane for which the following conditions are fulfilled?
- \latex{y\gt x^2}
- \latex{-x^2\lt y\lt x^2}
- \latex{\left| y\right| \leq x^2}
{{exercise_number}}. Plot the graph of the following function.
\latex{f:[1; +\infty[\rightarrow \R}, \latex{f(x)=\sqrt{x+2\sqrt{x-1}}+\sqrt{x-2\sqrt{x-1}}}
{{exercise_number}}. Determine the largest value of function \latex{ g } and the \latex{ x } at which the function takes this value.
\latex{g: \left]0; \frac{1}{2}\right]\rightarrow\R}, \latex{g(x)=\frac{1-\sqrt{1-4x^2}}{x}}
{{exercise_number}}. Determine the largest and the smallest value of the function \latex{ h }.
\latex{h: \left[0; 2\right]\rightarrow\R}, \latex{h(x)=\sqrt{2x-x^2}}