QuestionDetermine the domain, range, intervals of increase/decrease, x-intercepts, and y-intercept for $f(x)=2|x+3|-6$ .

Studdy Solution

STEP 1

Assumptions1. The function is given as $f(x)=|x+3|-6$ . We are looking for the domain, range, zones of increasing and decreasing, x-intercept(s), and y-intercept of the function.

STEP 2

First, let's find the domain of the function. The domain of a function is the set of all possible input values (x-values) which will produce a valid output. For this function, since absolute value is defined for all real numbers, the domain is all real numbers.
$Domain (-\infty, \infty)$

STEP 3

Next, let's find the range of the function. The range of a function is the set of all possible output values (y-values) that the function can produce. Since the absolute value function always produces non-negative results and we are subtracting6 from it, the range will be all numbers greater than or equal to -6.
$Range [-6, \infty)$

STEP 4

Now, let's find the zones of increasing and decreasing. The function $f(x)$ is increasing when the values of $f(x)$ increase as $x$ increases, and decreasing when the values of $f(x)$ decrease as $x$ increases.For the function $f(x)=2|x+3|-6$ , we can split it into two cases $f(x)=2(x+3)-6$ for $x\geq-3$ and $f(x)=-2(x+3)-6$ for $x<-3$ .

STEP 5

For $x\geq-3$ , the function $f(x)=2(x+3)-$ is a linear function with a positive slope (2), so it is increasing on the interval $[-3, \infty)$ .
For $x<-3$ , the function $f(x)=-2(x+3)-$ is a linear function with a negative slope (-2), so it is decreasing on the interval $(-\infty, -3)$ .
Therefore, the function is increasing on $[-3, \infty)$ and decreasing on $(-\infty, -3)$ .

STEP 6

Next, let's find the x-intercept(s). The x-intercepts of a function are the x-values where the function equals zero. To find the x-intercepts, we set $f(x)=0$ and solve for $x$ .
$0=2|x+3|-6$

STEP 7

olving for $x$ gives us $2|x+3|=6$ $|x+3|=3$

STEP 8

This gives us two solutions, $x+3=3$ and $x+3=-3$ .
olving these equations gives $x=0$ and $x=-6$ . So, the x-intercepts are $x=0$ and $x=-6$ .

STEP 9

Finally, let's find the y-intercept. The y-intercept of a function is the y-value where the function crosses the y-axis. This occurs when $x=$ .Substitute $x=$ into the function $f()=2|+3|-6$

STEP 10

olving for $f(0)$ gives us $f(0)=2(3)-6=0$ So, the y-intercept is $y=0$ .
In conclusion, the domain of the function is $(-\infty, \infty)$ , the range is $[-6, \infty)$ , the function is increasing on $[-3, \infty)$ and decreasing on $(-\infty, -3)$ , the x-intercepts are $x=0$ and $x=-6$ , and the y-intercept is $y=0$ .

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