Questions: Graph the function. y=-log3(x-2)+1 Specify the domain, range, intercept(s), and asymptote. (If an answer does not exist, enter DNE.) Domain: (-2, infty) (2, infty) (0, infty) (-infty, infty) (-2,4) Range: (2, infty) (-infty, 2) (0, infty) (-infty, infty) (-infty, 3) x-intercept (x, y)=()

Graph the function.
y=-log3(x-2)+1

Specify the domain, range, intercept(s), and asymptote. (If an answer does not exist, enter DNE.) Domain:
(-2, infty)
(2, infty)
(0, infty)
(-infty, infty)
(-2,4)

Range:
(2, infty)
(-infty, 2)
(0, infty)
(-infty, infty)
(-infty, 3)
x-intercept
(x, y)=()

Transcript text: Graph the function. \[ y=-\log _{3}(x-2)+1 \] Specify the domain, range, intercept(s), and asymptote. (If an answer does not exist, enter DNE.) Domain: $(-2, \infty)$ $(2, \infty)$ $(0, \infty)$ $(-\infty, \infty)$ $(-2,4)$ Range: $(2, \infty)$ $(-\infty, 2)$ $(0, \infty)$ $(-\infty, \infty)$ $(-\infty, 3)$ $x$-intercept $(x, y)=($ $\square$

Solution

Solution Steps

Step 1: Find the domain

The argument of the logarithm must be positive, so $x - 2 > 0$, which implies $x > 2$. Thus, the domain is $(2, \infty)$.

Step 2: Find the range

The range of a logarithmic function is all real numbers. Since the given function is a reflection and vertical shift of a logarithmic function, the range remains $(-\infty, \infty)$.

Step 3: Find the x-intercept

The x-intercept occurs when $y = 0$. So, $0 = -\log_3(x-2) + 1$. Then, $\log_3(x-2) = 1$. Rewriting in exponential form gives $x - 2 = 3^1 = 3$. Therefore, $x = 5$. The x-intercept is $(5, 0)$.