What are the domain and range of f(x) = |x – 3 | + 6? a. domain: {x | x is all real numbers} range: {y | y ≥ 6} b. domain: {x | x ≥ 3} range: {y | y ≥ 6} c. domain: {x | x is all real numbers} range: {y | y ≥ – 6} d. domain: {x | x ≥ 3} range: {y | y ≥ – 6}

a. Domain: {x | x is all real numbers} Range: {y | y ≥ 6}

Step-by-step explanation:

This is a modular function. All values inside brackets will result in absolute values. In other words, positive numbers.

And we can solve this problem algebraically and graphically.

f(x) = |x-3|+6

There are no restrictions and this function is continuous since we can proceed and take it out the brackets as we'll proceed.

I) y =|x-3|+6

Assuming the symmetrical values for x and -3, i.e. -x and +3

y=-(x-3)+6

y= -x+3+6

y=-x+9

We have a linear equation with no restriction in the Set of Real Numbers. We can choose any value for x, we'll have another for y.

II) Assuming the same values we did in the brackets.

y= x-3 +6

y=x+3

We have another linear equation with no restriction. We can choose any value for x, we'll have another for y.

Finally, we can affirm that the

Domain is  a number that belongs to this Interval: (-∞,∞) inside the Set of Real Numbers or

D={x∈R} even or just like the way it's been written {x|x is all Real Numbers}

Range

The range of function depends on the value set for x since it's the result of y.

But we can say where it starts. And in case of restrictions where it may not continue.

In this case, the range of the function starts at:

y=|x-3|+6

As the range starts at 6, precisely the point (0,6) no matter which value you pick for x. As the function continues with no restriction and goes on and on we can say the Range belongs to this interval [6, ∞) or {y ∈ R /y≥6}

Domain: {x | x is all real numbers} Range: {y | y ≥ 6}

Step-by-step explanation:

Domain is the set of all x values. No operations restrict the domain in this case. The domain is all real numbers.

Range is the set of all y values. Absolute value has a v shape starting at 0. Adding 6 raises this vertex to 6. The range is therefore all numbers greater than or equal to 6.