Question

Let f(x) = 1+ | x-1| , -2≤ x ≤ 2 find range and domain

Pls answer correctly

Answer 1

The domain is given in the question itself.

-2 ≤ x ≤ 2

For finding the range, let f(x) = y

y = 1 + |x - 1|

Now, for x < 1, we have

|x - 1| = 1 - x

y = 1 + 1 - x

→ y = 2 - x

→ x = 2 - y

Given that,

-2 ≤ x ≤ 2

Also, we assumed x < 1 at the beginning

so, for -2 ≤ x < 1, we have

x = 2 - y

Therefore, for x ≥ -2

2 - y ≥ -2

→ y ≤ 4

and for x < 1

→ 2 - y < 1

→ y > 1

So, here we got y is in the range (1, 4]

This is not the answer yet.

For x ≥ 1,

|x - 1| = x - 1

y = 1 + x - 1

→ y = x

Therefore, for x ≥ 1 and x ≤ 2, y = x

→ y ≥ 1 and y ≤ 2

So, the range in this part is [1, 2]

Hence, the range of f (x) will be union of the two intervals we found for y

That is, (1, 4] U [1,2]

Hence, range of f (x) = [1, 4]

Alternatively

You can just put value of x in f(x) as x is in a finite domain.

Put x = 2, we get y = 2

put x = 1, we get y = 1

put x = 0, we get 2

put x = -1, we get 3

put x = -2, we get 4

Hence, we can directly say the range of f(x) is [1, 4]

Answer 2

DOMAIN

The domain is the set of all possible x-values which will make the function "work", and will output real y-values.

When finding the domain, remember:

The denominator (bottom) of a fraction cannot be zero

The number under a square root sign must be positive.

TO FIND DOMAIN

In general, we determine the domain of each function by looking for those values of the independent variable (usually x) which we are allowed to use. (Usually we have to avoid 0 on the bottom of a fraction, or negative values under the square root sign).

RANGE

The range is the resulting y-values we get after substituting all the possible x-values.

TO FIND RANGE

The range of a function is the spread of possible y-values (minimum y-value to maximum y-value)

Substitute different x-values into the expression for y to see what is happening. (Ask yourself: Is y always positive? Always negative? Or maybe not equal to certain values?)

Make sure you look for minimum and maximum values of y.

Draw a sketch! In math, it's very true that a picture is worth a thousand words.

HOPE THIS HELPED YOU IN LEARNING TO FIND DOMAIN AND RANGE.

THANKS!