QuestionConsider a function $f(x)=-\sqrt{3x+a+2}$ .
c) In the space below, determine an expression representing the domain and range of $f(x)$ .
Do not provide the full form, provide the inequality portions only! For example, your domain could be $x<5$ and your range could be $y \geq-75$ .
D: $\square$
R: $\square$
d) In the space below, determine an expression representing the $x$ -intercept of $f(x)$ . state the $x$ -coordinate only. $x=$ $\square$

Studdy Solution

STEP 1

1. The function is $f(x) = -\sqrt{3x + a + 2}$ .
2. We need to find the domain and range of $f(x)$ .
3. We need to find the $x$ -intercept of $f(x)$ .

STEP 2

1. Determine the domain of $f(x)$ .
2. Determine the range of $f(x)$ .
3. Find the $x$ -intercept of $f(x)$ .

STEP 3

To find the domain, ensure the expression under the square root is non-negative: $3x + a + 2 \geq 0$ .

STEP 4

To find the range, consider the output of the function $f(x) = -\sqrt{3x + a + 2}$ . The square root function outputs non-negative values, so $-\sqrt{3x + a + 2}$ will output non-positive values.

STEP 5

To find the $x$ -intercept, set $f(x) = 0$ and solve for $x$ : $0 = -\sqrt{3x + a + 2}$ This implies $\sqrt{3x + a + 2} = 0$ .
The domain inequality is: $3x + a + 2 \geq 0$
The range inequality is: $y \leq 0$
The $x$ -intercept is found by solving: $3x + a + 2 = 0$

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