Question

Please help me solve this

Answer 1

The intervals expressed using set builder notation are;

(a) {x E R| -4 ≤ x < -5} and {x E R| 2.5 < x < 4.5}

(b) {x E R| -1.5 < x < 2.5}

(c) {x | x = 0}

(d) {x | x = 0}

(e) {x | x = 0}

(f) {x | x = 4.5}

What are set builder notations?

Set builder notation is a method used to express a set of numbers using curvy brackets in which the variable is defined, and possible values of the variable are indicated using inequality operators.

(a) The interval where f(x) is increasing are from x = -4 to x = -1.5, and from x = 2.5 to x < 4.5

Writing the intervals in set builder notation gives the following intervals;

{x E R| -4 ≤ x < -5} and {x E R| 2.5 < x < 4.5}

(b) The intervals where f(x) is decreasing are as follows;

f(x) decreases from x > -1.5 to x < 2.5

The interval in set builder notation is therefore;

{x E R| -1.5 < x < 2.5}

(c) f(x) = 0 at x = 0

The point where f(x) = 0 written in set builder notation is; {x | x = 0}

(d) From the graph, the value of f(0) is 0, given; f(x) = 0 at x = 0

The value of f(0) is 0, which gives;

{x | x = 0}

(e) From the graph, where f(x) = 0 is at x = 0

Which gives;

{x | x = 0}

(f) f(x) is undefined where f(x) does not have a value.

A point where f(x) does not have a value is indicated using an open circle, which is the point 4.5

Which gives;

f(x) is undefined at {x | x = 4.5}

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