Question

Study the function below and then answer the questions that follow. Equation What is the domain of f(x)? What is the range of f(x)?

Answer 1

Answer:

(a) Domain: All real numbers except -2

(b) $f(x) > -4$

Step-by-step explanation:

Given

See attachment for f(x)

Solving (a): The domain

From the attached image, we have:

$f(x) = -x - 2, x$

$f(x) = -x^2, -2 < x < 0$

$f(x) = x, x \ge 0$

To get the domain, we consider the inequalities attached to each of the piece-wise function

$x < -2$

This implies that the values of x is less than -2 i.e.

$x = \{.......,-4,-3\}$

$-2< x$

This implies that x is greater than -2 and less than 0 i.e.

$x = \{-1\}$

$x \ge 0$

This implies that x is greater than or equal to 0 i.e.

$x =\{0,1,2,3,....\}$

If these values of x are merged, we have:

$x = \{.......,-4,-3,-1,0,1,2,3.....\}$

$x = \{-\infty,..,-4,-3,-1,0,1,2,3,...,\infty\}$

-2 is not included in the above values of x.

Solving (b): The range

From the attached image, we have:

$f(x) = -x - 2, x$

$f(x) = -x^2, -2 < x < 0$

$f(x) = x, x \ge 0$

Substitute the greatest value of x in each piece-wise function.

$f(x) = -x - 2, x$

$x=-2$

So;

$f(-2) = -2-2 = -4$

$f(x) = -x^2, -2 < x < 0$

$x = -2$

So:

$f(-2) = -2^2 = -4$

$f(x) = x, x \ge 0$

$x = \infty$

So;

$f(\infty) = \infty$

We have:

$f(-2) =-4$

$f(-2) =-4$

$f(\infty) = \infty$

The smallest value of f(x) is -4

Hence, the range is:

$f(x) > -4$