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). 

Illustration Questions

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

Illustration Questions

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

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.

Illustration Questions

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.

Illustration Questions

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

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.

Illustration Questions

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.

Illustration Questions

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

Practice Now