Concept: Function & Graphs

CONTENTS

INTRODUCTION

Functions and Graphs is a very important chapter from CAT and XAT perspective. You can expect 2 to 3 quesetions from this chapter in CAT. SNAP and NMAT do not ask questions from this chapter.

FUNCTIONS

A function y = f(x) is where x is the input and f(x) is the output.

x is the independent variable here and y is the dependent variable.

Example: f(x) = 2x + 3 is a function whose input is x and (2x + 3) is the output.
If we have to calculate value of f(2), it means we need to calculate the output when input i.e., x = 2. ∴ f(2) = 2 × 2 + 3 = 7.

Similarly, if y = f(x) = x2 - 2x + 3, then
f(-1) = (-1)2 - 2 × (-1) + 3 = 1 + 2 + 3 = 6.

A function is defined only if for every input (x) there is only one output (f(x) or y).

y = 3x3 + 3x + 5 is a function, but

y2 = 2x + 3 is not a function.
because at x = 3, y can be +3 or -3 i.e., there are 2 outputs possible for the same input.

Composite Function

The composition of a function is an operation where two functions say f and g generate a new function say h in such a way that h(x) = g(f(x)). It means here function g is applied to the function of x. So, basically, a function is applied to the result of another function.

Example : Example: If f(x) = 2x + 3 and g(x) = x2, then find f(g(x))

Solution:
f(x) = 2x + 3

⇒ f(g(x)) = 2 × g(x) + 3

⇒ f(g(x)) = 2 × x2 + 3

Example : Example: If f(x) = 3x - 1 and g(x) = x2 + 1, then find f(g(1))

Solution:
g(x) = x2 + 1
⇒ g(1) = 12 + 1 = 2

Now, f(x) = 3x + 1
⇒ f(g(1)) = f(2) = 3 × 2 + 1 = 7

Note: f(g(x)) can also be written as fog(x). This means f or g(x).

Example : Example: If f(x) = 3x - 1, then find fofof(1)

Solution:
f(x) = 3x - 1
⇒ f(1) = 3 × 1 - 1 = 2

Now, fof(1) = f(2) = 3 × 2 - 1 = 5.

Now, fofof(1) = f(5) = 3 × 5 - 1 = 14.

Example : Example: If f(x) = 3x - 1 and g(x) = x2, then find fogof(0)

Solution:
f(x) = 3x - 1
⇒ f(0) = 3 × 0 - 1 = -1

Now, gof(0) = g(-1) = (-1)2 = 1.

Now, fogof(0) = f(1) = 3 × 1 - 1 = 2.

Piecewise Function

A function which has different definitions depending upon the value of the independent variable is called a Piecewise Function. Thus, the definition of the function changes according to the input.

Example: $\mathrm{f\left(x\right)}=\left\{\begin{array}{ll}\mathrm{x}& \mathrm{x}\ge 0\\ -\mathrm{x}& \mathrm{x}<0\end{array}\right\$

Here, if x is ≥ 0 f(x) = x and
if x < 0 , f(x) = -x

∴ f(2) = 2 but f(-2) = -(-2) = 2

Peiodic Function

A periodic function is a function that repeats itself after regular intervals. Thus, if f(x) = f(x + c), where c is some constant, for all values of x, then f(x) is a periodic function. c is called the period/periodicity of the function.

The most common periodic functions are the trigonometric functions like sin x, cos x etc.

Example : 𝑔(𝑥) = 𝑔(𝑥 + 1)+ 𝑔(𝑥 −1) for all 𝑥. For which value of 𝑝 is the relation 𝑔(𝑥 +𝑝) = 𝑔(𝑥) necessarily true for all 𝑥?

Solution:
𝑔(𝑥) = 𝑔(𝑥 + 1)+ 𝑔(𝑥 −1)

⇒ g(x + 1) = g(x) - g(x - 1)

Let g(1) = a and g(2) = b

⇒ g(3) = g(2) - g(1) = b - a

⇒ g(4) = g(3) - g(2) = b - a - b = -a

⇒ g(5) = g(4) - g(3) = -a - (b - a) = -b

⇒ g(6) = g(5) - g(4) = -b - (-a) = a - b

⇒ g(7) = g(6) - g(5) = a - b - (-b) = a

⇒ g(8) = g(7) - g(6) = a - (a - b) = b

We can see that g(7) = g(1), g(8) = g(2) and so on

∴ g(x + 6) = g(x)

g(x) is a periodic function whose periodicity is 6.

Even Odd Function

A function f(x) is an even if f(-x) = f(x)

Exmaple: f(x) = 22
Here, f(-x) = (-x)2 = x2 = f(x)

A function f(x) is a odd if f(-x) = -f(x)

Exmaple: f(x) = 23
Here, f(-x) = (-x)3 = -x3 = -f(x)

Note: Not all functions are even or odd. Some function may neither be even nor odd.

Inverse Function

If f(x) = y, then inverse function is defined as f-1y = x.

In a function y = f(x), x is the input and y is the output,
In an inverse function x = f-1y, y is the input and x is the output.

Example: If f(x) = y = 2x + 3, then
⇒ x = $\frac{\left(y - 3\right)}{2}$
⇒ f-1y = g(y) = $\frac{\left(y - 3\right)}{2}$

We can replace y with x, and hence the inverse function g(x) = $\frac{\left(y - 3\right)}{2}$

Not all functions have an inverse. Whether or not an inverse exists for a function, can be found using graphs. If the reflection of the graph of a function about the line y = x satisfies the vertical line test, then the inverse exists.

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