Informative line

### Definition And Value Of Function

Learn definition of a function with piecewise defined function formula and practice problems of graphing absolute value & linear functions with examples & equations.

# Definition and Value of a Function

Whenever one quantity depends on the other we say that first quantity is a function of the second. So basically we can say

(depends on ⇒ function of) in Calculus

For example the area of a square depends on the length of its sides, so we say

$$A = \ell^2$$ (Where A is the area of square and $$\ell$$ is the length of its side)

We will say Area is a function of length of sides .

$$A(\ell) = \ell^2$$

$$\ell$$ will be called the independent variable and A the dependent variable.

• So function is a rule that assigns to each element 'x' in a set A a unique element in set B (called f(x) or y). Note that for every x there should be only one f(x).
• Set A and B are usually the sets of real numbers.
• Suppose        f(x) = some expression or formula in x ,then to obtain f(a) where 'a' is a fixed value put the value of 'a' at all places where x occurs in the expression. The resultant value is f(a).

#### Consider the volume of a sphere which depends on its radius. $$V (r) = {4 \over 3} \pi r^3$$ (V indicates volume and r the radius) Which of the following is correct?

A Volume of sphere is not a function of its radius

B Volume of sphere is a function of its radius

C Volume of sphere will not change when radius change

D An increase in radius of sphere will not change its volume

×

Since the volume depends on radius ⇒ Volume is a function of radius

Note :depends on ⇒ function of

### Consider the volume of a sphere which depends on its radius. $$V (r) = {4 \over 3} \pi r^3$$ (V indicates volume and r the radius) Which of the following is correct?

A

Volume of sphere is not a function of its radius

.

B

Volume of sphere is a function of its radius

C

Volume of sphere will not change when radius change

D

An increase in radius of sphere will not change its volume

Option B is Correct

#### Let $$F(x) = x^2$$ for all real x, then find the value of f(2).

A –6

B –7

C 0

D 4

×

$$F(x) = x^2$$   → given formula

put x = 2 on both sides

$$f(2) = 2^2$$

f(2) = 4

### Let $$F(x) = x^2$$ for all real x, then find the value of f(2).

A

–6

.

B

–7

C

0

D

4

Option D is Correct

# Absolute Value Function

It is a piecewise defined function.

$$f(x)=|x|=\begin{cases} x & if\,\, x\geq0\\ -x & if \, x <0\end{cases}$$

is called absolute value function. It gives us the distance of x from the origin which is always positive.

|–2| = 2,  |3| = 3.

Note that the graph follows  $$y = x$$ to the right of origin and  $$y = –x$$ to the left of origin.

#### If f(x) = |2x – 1| then the value of f(2) is

A 5

B –10

C –7

D 3

×

f(x) = |2x – 1|

put x = 2 on both sides

f(2) = |2 × 2 – 1|

f(2) = |3| = 3

If the value inside the absolute value bracket is positive we return the value, if it is negative we make the sign positive.

### If f(x) = |2x – 1| then the value of f(2) is

A

5

.

B

–10

C

–7

D

3

Option D is Correct

# Odd and Even Functions

• If a function 'f' satisfies

$$f(–x) = f(x)$$  for every x in its domain then we say that it is an even function whereas

if

$$f(–x) = –f(x)$$ for all x in its domain we say that it is an odd function.

(1)  $$f(x) = x^2$$ is an example of an even function, as

$$f(-x) = (-x)^2 = (-1)^2 \times x^2 = 1 \times x^2 = x^2$$

$$= f(x)$$

(2)  $$f(x) = x^3$$ is an example of an odd function as

$$f(-x) = (-x)^3 = (-1)^3 x^3 = -1 \times x^3 = -x^3$$

$$= –f(x)$$

• To test whether a function is odd or even we apply the definition. Also note that there are functions which are neither odd nor even.

example: $$f(x) = x + x^2$$ is neither odd nor even.

#### Let $$f(x) = 1 - x^4$$ then  $$f(x)$$ is

A odd

B even

C neither odd nor even

D nothing can be said

×

$$f(x) = 1- x^4$$

Replace x by –x on both sides

$$f(-x) = 1 - (-x)^4$$

$$f(-x) = 1 - (-1)^4 x^4$$

$$f(-x) = 1 - x^4 = f(x)$$

### Let $$f(x) = 1 - x^4$$ then  $$f(x)$$ is

A

odd

.

B

even

C

neither odd nor even

D

nothing can be said

Option B is Correct

# Linear Function and its Sign

• A function of the form   $$f(x)=ax+b$$  is called a linear function (its graph represents a straight line, therefore the name).
• This expression takes positive, negative or zero values depending or what values of  $$x$$  we take.

#### Consider $$f(x)=x-8$$, what is the sign of this expression for $$x=-9$$ .

A Negative

B Positive

C 0

D Nothing can be said

×

$$f(x)=x-8$$

Put $$x=-9$$ in the expression on both sides.

$$f(-9)=-9-8$$

$$f(-9)=-17\rightarrow$$So the expression is negative.

### Consider $$f(x)=x-8$$, what is the sign of this expression for $$x=-9$$ .

A

Negative

.

B

Positive

C

0

D

Nothing can be said

Option A is Correct

# Piecewise Defined Functions

• Piecewise defined functions are those which are defined by different formulas in different parts of their domain.
• Consider

$$f(x)=\begin{cases} 2x+1 & if\,\, x\leq 1\\ \ x & if \,\, x >1 \end{cases}$$

is a piecewise defined function. For all values of x greater than 1, it is given by x and for those less than or equal to 1, it is given by 2x + 1.

#### A function f is defined by $$f(x)=\begin{cases} 2x^2+1 & if\,\, x\leq -1\rightarrow 1^{st}rule\\ \ 2x+1 & if \,\, x >1\rightarrow\,2^{nd} rule \end{cases}$$ find the value of f(2).

A 1

B 3

C 5

D 9

×

Since 2 is a value greater than –1 so for f(2) we apply  $$2^{nd} \space rule$$

f(2) = 2(2) + 1

= 4 + 1 = 5

### A function f is defined by $$f(x)=\begin{cases} 2x^2+1 & if\,\, x\leq -1\rightarrow 1^{st}rule\\ \ 2x+1 & if \,\, x >1\rightarrow\,2^{nd} rule \end{cases}$$ find the value of f(2).

A

1

.

B

3

C

5

D

9

Option C is Correct