## Functions, Graphs and Statistics

Theory

This topic generally does not have much weightage in CAT. The problems from this topic can be deceptively difficult – imposing when first read but with simple solutions. Statistics is generally straightforward and these questions of mean, median, mode should not be missed. Once you understand the concept of reflection and shifting of curves, the questions on those concepts would become very straightforward.

Theory

Domain and Range: The domain of a real function f are the range of real values that return a real output. For example: $$f$$ = $$\sqrt{16-x^2}$$. The function f will have real values only for $$x^2 \leq 16$$. Hence, $$-4 \leq x \leq 4$$. Hence, the domain of f is [-4,4] ([ ] square brackets imply that the domain has all values between the two boundaries including the boundary values, in this case including 4 and -4. ( ) round brackets imply all values between -4 and 4 excluding them.) The Range of a real function f is the range of real values that can be returned by a function. For eg. $$f = \left | x + 4 \right | - \left | x - 4 \right |$$. For x>=4, the value of the function is constant at +8. Similarly for $$x \leq -4$$, the output is constant at -8. Between -4 to +4 the output varies between -8 to +8. Hence, the range of possible outputs is [-8,8].

Theory

Even and odd function: A function f is an even function if f(-x)=f(x). For eg. $$f(x)=x^4$$ + $$x^2$$. In this case $$f(-x)= (-x)^4$$ + $$(-x)^2=x^4$$ + $$x^2=f(x)$$. Hence f(x) is an even function. A function f is an odd function if f(-x)=-f(x). For eg. $$f(x)=x^3$$ + $$x$$. In this case $$f(-x)= (-x)^3$$ + $$(-x)=-(x^3$$ + $$x)= -f(x)$$. Hence f(x) is an odd function.

Theory

Graphs: Drawing common graphs

Consider the function Y= | X |. Hence for $$X \geq 0$$, Y=X and for $$X \leq 0$$, Y=-X. This can be drawn as two lines with slopes 1 and -1 meeting at the origin. Similarly the function $$X^2$$ + $$Y^2=R^2$$ is a circle with the radius R. The function $$Y=X^2$$ is a parabola symmetric around the Y-axis. To draw a graph, plot a few points. If the equation is of the power=1, connect them with a straight line. Else, join them with a curve. The image shows the graphs of commonly occurring questions.

Shifting a curve: To draw a curve f(x-r) we shift f(x) r units to the right. Similarly, to draw the curve f(x+r) we shift f(x) r units to the left. The bottom left panel of the image shows y=| x –r |. To draw this curve, draw y=|x| and move the curve r units to the right as shown in the image.

Reflection in x and y axes: Let f(x)=| x –r | to draw –f(x) we have to reflect the curve in the x axis as shown in the image in the bottom right panel. To draw f(-x) we should reflect the curve around the y axis.

Theory

Statistics:

Mean : This is same as average of the numbers. For eg mean of S={1,2,3,5,9} is 4.

Median : When the set of numbers are arranged in an ordered list, the median is the middle number. In case the set of numbers have even number of numbers then the median is the average of the middle two numbers. For eg median of S={1,2,3,5,9} is 3.

Mode : Mode is the most often repeated number in a set. For eg. S={1,3,3,5,9} has mode = 3.