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

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).

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

- 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.

A Negative

B Positive

C 0

D Nothing can be said

- 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.

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 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.