Question

The graph represents the piecewise function:

f(x) =

What is the domain and range of the function

Answer 1

Let us examine the graph.

When x = 2, y = 2

f(2) = 2

When x = - 2, y = 2

f(-2) = 2

When x = 0, y = 0

f(0) = 0

From this we can conclude that, The graph is showing a function which gives the absolute value of a number.

The function can thus be,

f(x) = | x |

This piece wise function is called The Absolute value function, or The Modulo function.

It is defined as,

f(x) = x, if x > 0

f(x) = 0, if x = 0

f(x) = - x if x < 0.

The Domain of this function is The Real Numbers

The Range of this function is From Zero to positive infinite.

Domain : R

$\bold{Range : [ \: \: 0 \: , \infty )}$

Answer 2

The Domain of this function is all the Real Numbers, Domain : R

The Range of this function is From Zero to positive infinite, Range : [0, ∞)

• We must note some co-ordinates of the graph:

x₁ = 0 , y₁ = 0

x₂ = ±1 , y₂ = 1

x₃ = ±2 , y₃ = 2

x₄ = ±3 , y₄ = 3

The co-ordinates help us identify the relationship between x and and the f(x) function.

• Note that all the positive and negative values of x co-ordinates have their respective modulus functions as their y co-ordinates. For example

⇒ $x = \pm1$

⇒ $\lvert \ x\ \rvert = \lvert\ \pm1\ \rvert = 1$

So,

⇒ $y = 1$

⇒ $y=f(x) = \lvert \ x \ \rvert$

• The Domain of a function is all values which are input in the function and offer a definite vale as output.
• For, $f(x) = \lvert \ x \ \rvert$, modulus function is put to use. Modulus function changes negative sign of a value to positive sign thus all Real numbers can be input into any modulus function.

Domain: Real Numbers, R

• Range of a function refers to all the possible values of the output of the function.
• For a modulus function, because the domain is All real numbers, the range will result to be from 0 to positive infinite. All values as the output of the modulus function are positive.

Range: [0, ∞)

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