QuestionIdentify the true characteristics of the function $f(x)=-|x+2|-3$ . Options include domain, range, and behavior as $x$ approaches infinity.

Studdy Solution

STEP 1

Assumptions1. The function is $f(x)=-|x+|-3$ . We are looking for the characteristics of the function that are true.

STEP 2

Let's first understand the function. The function is a transformation of the absolute value function. The absolute value function $|x|$ is reflected about the x-axis, shifted2 units to the left and units down.

STEP 3

The domain of a function is the set of all possible x-values which will make the function "work", and will output real y-values. Since there are no restrictions on x in the given function, the domain of the function is all real numbers.

STEP 4

The range of a function is the set of all possible output values (y-values), which result from using the function formula. The function $f(x)=-|x+2|-3$ will always be less than or equal to -3 because the absolute value function is always non-negative and we are subtracting it from -3. So, the range of the function is $(-\infty, -3]$ .

STEP 5

The function is increasing when the values of $x$ are getting larger. To find where the function is increasing, we can look at the derivative of the function. However, since this is an absolute value function, we can also look at the graph or use the properties of absolute value functions. The function $f(x)=-|x+2|-3$ is increasing when $x<-2$ .

STEP 6

The function is decreasing when the values of $x$ are getting smaller. The function $f(x)=-|x+2|-3$ is decreasing when $x>-2$ .

STEP 7

As $x \rightarrow \infty$ , the function $f(x)=-|x+2|-3$ does not approach $\infty$ . Instead, it approaches $-\infty$ . Similarly, as $x \rightarrow -\infty$ , the function does not approach $\infty$ , it approaches $-\infty$ .

STEP 8

The function $f(x)=-|x+2|-3$ is negative for all real values of $x$ because the absolute value function is always non-negative and we are subtracting it from -3, which will always result in a negative number.

STEP 9

Now, let's compare our findings with the given optionsa. The domain is all real numbers, but the range is not. b. The function is increasing when $x<-2$ , not when $x \leq-3$ . c. The function is decreasing when $x>-2$ , not when $x \geq-2$ . d. As $x \rightarrow \infty, f(x) \rightarrow -\infty$ and as $x \rightarrow-\infty, f(x) \rightarrow -\infty$ . e. The function is negative for all real numbers.
So, the only correct characteristics are (e) The function is negative for all real numbers.

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