Questions: Select the correct answer from each drop-down menu. The x-intercept, or zero, of function g is x= Function g is over the interval [-5,5]. As the value of x approaches positive infinity, the value of g(x) approaches infinity.

Select the correct answer from each drop-down menu.

The x-intercept, or zero, of function g is x=
Function g is over the interval [-5,5].

As the value of x approaches positive infinity, the value of g(x) approaches infinity.

Transcript text: Select the correct answer from each drop-down menu. The $x$-intercept, or zero, of function $g$ is $x=$ $\square$ Function $g$ is $\square$ over the interval $[-5,5]$. As the value of $x$ approaches positive infinity, the value of $g(x)$ approaches $\square$ infinity.

Solution

Solution Steps

Step 1: Determine the x-intercept of the function g(x)

To find the x-intercept, set $ g(x) = 0 $ and solve for $ x $: \[ 0 = \sqrt[3]{x - 3} \] Cubing both sides: \[ 0 = x - 3 \] So, \[ x = 3 \]

Step 2: Determine if the function g is increasing or decreasing over the interval [-5, 5]

To determine if the function is increasing or decreasing, observe the graph of $ g(x) = \sqrt[3]{x - 3} $. The function is decreasing as $ x $ increases from -5 to 5.

Step 3: Determine the behavior of g(x) as x approaches positive infinity

As $ x $ approaches positive infinity, $ g(x) = \sqrt[3]{x - 3} $ also approaches positive infinity because the cube root function grows without bound as its argument increases.

Final Answer

The x-intercept, or zero, of function g is $ x = 3 $.
Function g is decreasing over the interval $[-5, 5]$.
As the value of $ x $ approaches positive infinity, the value of $ g(x) $ approaches positive infinity.

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