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Power functions and root functions
Reminder
We became familiar with the quadratic function in year \latex{9}. The image of the function
\latex{f:\R\to\R}, \latex{f\left(x\right)=x^{2}}
is the so called standard parabola. (Figure 1)
The inverse of the function \latex{f:\R^{+}_{0}\to \R}, \latex{f\left(x\right)=x^{2}} is the function below (Figure 1):
\latex{g:\R^{+}_{0}\to \R}, \latex{g\left(x\right)=\sqrt{x}}.
It means that at any non-negative place \latex{x}
\latex{g\left(f\left(x\right) \right)=x }, since \latex{\sqrt{x^2}=\mid x \mid=x}, if \latex{x \geq 0}.
Figure 1
\latex{y=x^2}
\latex{y=x}
\latex{y=\sqrt{x}}
\latex{ 1 }
\latex{ 1 }
\latex{ y }
\latex{ x }
This characteristic can be seen in the graph of the functions so that the mirror image of the graph of the function \latex{f(x) = x^2} \latex{(x \geq 0)} about the straight line \latex{y = x} is the graph of the function \latex{g(x)= \sqrt x}.
Example 1
Let us plot the graphs of and characterise the following functions:
\latex{f:\R\to\R}, \latex{f(x)=x^{3}} and \latex{g:\R\to\R}, \latex{g(x)=\sqrt[3]{x}}.
Solution
By calculating the values of \latex{f} at a few places the following curve can be drawn (Figure 2). The range of the function \latex{f(x) = x^3} is the set of real numbers; it is an increasing and odd function; it is concave for negative places \latex{x}, it is convex for positive places \latex{x}. Since it takes every value only at one place, it has an inverse.
We know that \latex{\sqrt[3]{x^3}=x}, therefore the inverse of the function \latex{f(x) = x^3} is the function \latex{g: \R \to \R, g(x) = \sqrt[3]{x}}.
The graph of the function \latex{g(x)=\sqrt[3]{x}} can be derived from the graph of the function \latex{f(x) = x^3} by reflecting it about the straight line \latex{y = x}. (Figure 2)
The range of the function \latex{g(x) =\sqrt[3]{x}} is the set of real numbers; it is an increasing function; it is concave for positive places \latex{x}, it is convex for negative places \latex{x}.
The graph of the function \latex{g(x)=\sqrt[3]{x}} can be derived from the graph of the function \latex{f(x) = x^3} by reflecting it about the straight line \latex{y = x}. (Figure 2)
The range of the function \latex{g(x) =\sqrt[3]{x}} is the set of real numbers; it is an increasing function; it is concave for positive places \latex{x}, it is convex for negative places \latex{x}.
Figure 2
\latex{y=x^3}
\latex{y=x}
\latex{y=\sqrt[3]{x}}
\latex{ 1 }
\latex{ 1 }
\latex{ y }
\latex{ x }
Additional functions
Example 2
Let us plot the graphs of the following functions:
\latex{f:\R\to\R}, \latex{f(x)=x^{4}} and \latex{g:\R^{+}_{0}\to\R}, \latex{g(x)=\sqrt[4]{x}}.
Solution
After calculating a few substitution values the graph of the function \latex{f} can be plotted. A curve similar to the parabola results, however the graph of our function is still not a parabola, since if \latex{x \lt –1} or \latex{x \gt 1}, then a curve “steeper” than the parabola results, and if \latex{–1 \lt x \lt 1}, then a curve “shallower” than the parabola results. (Figure 3)
Our function is not one-to-one, therefore it does not have an inverse function. If we restrict the domain to the set of non-negative numbers, then we obtain a one-to-one function, which has an inverse.
Our function is not one-to-one, therefore it does not have an inverse function. If we restrict the domain to the set of non-negative numbers, then we obtain a one-to-one function, which has an inverse.
Since in the case of non-negative numbers \latex{\sqrt[4]{x^{4} }=x}, the inverse of the function \latex{f_1:\R^{+}_{0}\to\R }, \latex{f_1(x)=x^4} is the function \latex{g:\R^{+}_{0}\to\R }, \latex{g(x)=\sqrt[4]{x}}.
The graph of \latex{g} can be derived from the graph of \latex{f_{1}} by reflecting it about the straight line \latex{y = x}. (Figure 3)
The graph of \latex{g} can be derived from the graph of \latex{f_{1}} by reflecting it about the straight line \latex{y = x}. (Figure 3)
Figure 3
\latex{y=x^4}
\latex{y=x}
\latex{y=\sqrt[4]{x} }
\latex{ 1 }
\latex{ 1 }
\latex{ y }
\latex{ x }
Example 3
Let us plot the graphs of and characterise the following functions, the domain of which is the suitable subset of the set of real numbers, and the codomain of which is the set of real numbers.
- \latex{f(x)=\frac{1}{2}\times\left(x-3\right)^3-2} \latex{\left(x\in \R\right)}
- \latex{g(x)=-\sqrt[4]{x+2}+3 } \latex{\left(x\geq-2\right) }
Solution (a)
The function \latex{f_1(x)=(x – 3)^3} takes the same value at a place \latex{3} greater as the function \latex{x \to x^{3}} does. Therefore its graph is derived by translating the image of the function \latex{x \to x^{3}} by \latex{3} units to the positive direction along the \latex{x}-axis. (Figure 4)
The image of the function \latex{f_2(x)=\frac{1}{2}\times(x-3)^3} is derived from the graph of \latex{f_1} by reducing the \latex{y}-coordinate of every point of the curve to \latex{\frac{1}{2}} its value. (Figure 4)
The image of the function \latex{f_2(x)=\frac{1}{2}\times(x-3)^3} is derived from the graph of \latex{f_1} by reducing the \latex{y}-coordinate of every point of the curve to \latex{\frac{1}{2}} its value. (Figure 4)
The image of the function \latex{f} can be derived from the graph of the function \latex{f_2} by translating it by \latex{2} downwards, to the negative direction along the \latex{y}-axis. (Figure 4)
The range of the function \latex{f} is the set of real numbers; it is an increasing function; it is concave if \latex{x \lt 3}, it is convex if \latex{x \gt 3}.
The range of the function \latex{f} is the set of real numbers; it is an increasing function; it is concave if \latex{x \lt 3}, it is convex if \latex{x \gt 3}.
Figure 4
\latex{y=x^3}
\latex{y=\frac{1}{2}\times \left(x-3\right)^3}
\latex{y=\frac{1}{2}\times \left(x-3\right)^3-2}
\latex{y=\left(x-3\right)^3}
\latex{ 1 }
\latex{ 1 }
\latex{ 3 }
\latex{ 2 }
y
x
Figure 5
\latex{y=\sqrt[4]{x+2}}
\latex{y=-\sqrt[4]{x+2}+3}
\latex{y=\sqrt[4]{x}}
\latex{y=-\sqrt[4]{x+2}}
\latex{ 1 }
\latex{ 1 }
\latex{ -2 }
y
x
Solution (b)
The image of the function \latex{g_1(x) =\sqrt[4]{x+2}} is derived by translating the graph of the function \latex{x\to \sqrt[4]{x}} along the \latex{x}-axis by \latex{2} to the negative direction. (Figure 5)
The image of the function \latex{g_2(x) =-\sqrt[4]{x+2}} is derived from the graph of \latex{g_1} by reflecting it about the \latex{x}-axis. (Figure 5)
The image of the function \latex{g_2(x) =-\sqrt[4]{x+2}} is derived from the graph of \latex{g_1} by reflecting it about the \latex{x}-axis. (Figure 5)
To sketch the graph of the function \latex{g} we have to translate the graph of the function \latex{g_2} by \latex{3} along the \latex{y}-axis upwards to the positive direction. (Figure 5)
The range of the function g is the set of real numbers not greater than \latex{3}; at the value \latex{x = –2} it has a maximum; its maximum value is \latex{3}. It is a decreasing function on the interval \latex{[–2; \infty[}; it is convex on this interval.
The range of the function g is the set of real numbers not greater than \latex{3}; at the value \latex{x = –2} it has a maximum; its maximum value is \latex{3}. It is a decreasing function on the interval \latex{[–2; \infty[}; it is convex on this interval.
Exercises
{{exercise_number}}. Give the largest subset of the set of real numbers on which the following expressions have a meaning. With the help of function transformations sketch the graphs of the functions on the subset of the set of real numbers on which they have a meaning.
- \latex{f(x) =\sqrt[3]{x-5}}
- \latex{f(x)=\left(x+3\right)^4}
- \latex{f(x) =\sqrt[4]{x+4}}
- \latex{f(x)=\left(x-2\right)^3-4}
- \latex{f(x) =\sqrt[4]{x-3}-2}
- \latex{f(x) =\sqrt[3]{x+2}+4}
- \latex{f(x)=-\left(x-4\right)^4+2}
- \latex{f(x) =\sqrt[4]{3-x}+4}
- \latex{f(x) =3\times \sqrt[3]{3x}-3}
Puzzle
Determine the zeros of the functions in example 3.