Math Snap

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.

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:
$$3x + a + 2 = 0$$