Answer :
To analyze the function [tex]\( g(x) \)[/tex], let's carefully examine its properties on both pieces of its definition.
### Step 1: Determine the [tex]\( x \)[/tex]-intercepts
The [tex]\( x \)[/tex]-intercepts of a function are the points where the function crosses the x-axis, i.e., where [tex]\( g(x) = 0 \)[/tex].
For [tex]\( x < 0 \)[/tex]:
[tex]\[ g(x) = \left(\frac{3}{4}\right)^2 \][/tex]
This is a constant positive value, so there are no [tex]\( x \)[/tex]-intercepts in this region.
For [tex]\( x \geq 0 \)[/tex]:
[tex]\[ g(x) = -x^2 \][/tex]
This quadratic function is zero when [tex]\( x = 0 \)[/tex]. Thus, there is one [tex]\( x \)[/tex]-intercept at [tex]\( x = 0 \)[/tex].
So, function [tex]\( g \)[/tex] has 1 [tex]\( x \)[/tex]-intercept.
### Step 2: Determine the [tex]\( y \)[/tex]-intercepts
The [tex]\( y \)[/tex]-intercepts of a function are the points where the function crosses the y-axis, i.e., where [tex]\( x = 0 \)[/tex].
Evaluating the function at [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = -0^2 = 0 \][/tex]
So, the function crosses the y-axis at [tex]\( y = 0 \)[/tex].
So, function [tex]\( g \)[/tex] has 1 [tex]\( y \)[/tex]-intercept.
### Step 3: Determine whether the function is decreasing
A function is decreasing if it gets smaller as [tex]\( x \)[/tex] increases.
For [tex]\( x < 0 \)[/tex]:
[tex]\[ g(x) = \left(\frac{3}{4}\right)^2 \][/tex]
This is constant and does not decrease, but it is not really relevant for decreasing behavior.
For [tex]\( x \geq 0 \)[/tex]:
[tex]\[ g(x) = -x^2 \][/tex]
This quadratic function opens downwards, meaning [tex]\( g(x) \)[/tex] decreases as [tex]\( x \)[/tex] increases from [tex]\( x = 0 \)[/tex].
Thus, function [tex]\( g \)[/tex] is decreasing on its effective interval [tex]\( x \geq 0 \)[/tex].
### Step 4: Determine whether the function is continuous
A function is continuous if there are no breaks, jumps, or holes in its graph.
For [tex]\( x < 0 \)[/tex]:
[tex]\[ g(x) = \left(\frac{3}{4}\right)^2 \][/tex]
This is a continuous constant function.
For [tex]\( x \geq 0 \)[/tex]:
[tex]\[ g(x) = -x^2 \][/tex]
This is a continuous polynomial function.
However, we need to examine the point [tex]\( x = 0 \)[/tex] to check for continuity:
- From the left, as [tex]\( x \)[/tex] approaches 0, [tex]\( g(x) = \left(\frac{3}{4}\right)^2 \)[/tex].
- From the right, as [tex]\( x \)[/tex] approaches 0, [tex]\( g(0) = 0 \)[/tex].
The left and right limits at [tex]\( x = 0 \)[/tex] are not equal (since [tex]\( \left(\frac{3}{4}\right)^2 \neq 0 \)[/tex]). Thus, there is a discontinuity at [tex]\( x = 0 \)[/tex].
So, function [tex]\( g \)[/tex] is not continuous.
### Summary
Function [tex]\( g \)[/tex] has:
- 1 [tex]\( x \)[/tex]-intercept.
- 1 [tex]\( y \)[/tex]-intercept.
- The function is decreasing on its effective interval.
- The function is not continuous.
Thus, the correct answers to fill in the blanks are:
1. [tex]\( 1 \)[/tex] for [tex]\( x \)[/tex]-intercepts.
2. [tex]\( 1 \)[/tex] for [tex]\( y \)[/tex]-intercepts.
3. "is" for decreasing.
4. "not" for continuous.
### Step 1: Determine the [tex]\( x \)[/tex]-intercepts
The [tex]\( x \)[/tex]-intercepts of a function are the points where the function crosses the x-axis, i.e., where [tex]\( g(x) = 0 \)[/tex].
For [tex]\( x < 0 \)[/tex]:
[tex]\[ g(x) = \left(\frac{3}{4}\right)^2 \][/tex]
This is a constant positive value, so there are no [tex]\( x \)[/tex]-intercepts in this region.
For [tex]\( x \geq 0 \)[/tex]:
[tex]\[ g(x) = -x^2 \][/tex]
This quadratic function is zero when [tex]\( x = 0 \)[/tex]. Thus, there is one [tex]\( x \)[/tex]-intercept at [tex]\( x = 0 \)[/tex].
So, function [tex]\( g \)[/tex] has 1 [tex]\( x \)[/tex]-intercept.
### Step 2: Determine the [tex]\( y \)[/tex]-intercepts
The [tex]\( y \)[/tex]-intercepts of a function are the points where the function crosses the y-axis, i.e., where [tex]\( x = 0 \)[/tex].
Evaluating the function at [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = -0^2 = 0 \][/tex]
So, the function crosses the y-axis at [tex]\( y = 0 \)[/tex].
So, function [tex]\( g \)[/tex] has 1 [tex]\( y \)[/tex]-intercept.
### Step 3: Determine whether the function is decreasing
A function is decreasing if it gets smaller as [tex]\( x \)[/tex] increases.
For [tex]\( x < 0 \)[/tex]:
[tex]\[ g(x) = \left(\frac{3}{4}\right)^2 \][/tex]
This is constant and does not decrease, but it is not really relevant for decreasing behavior.
For [tex]\( x \geq 0 \)[/tex]:
[tex]\[ g(x) = -x^2 \][/tex]
This quadratic function opens downwards, meaning [tex]\( g(x) \)[/tex] decreases as [tex]\( x \)[/tex] increases from [tex]\( x = 0 \)[/tex].
Thus, function [tex]\( g \)[/tex] is decreasing on its effective interval [tex]\( x \geq 0 \)[/tex].
### Step 4: Determine whether the function is continuous
A function is continuous if there are no breaks, jumps, or holes in its graph.
For [tex]\( x < 0 \)[/tex]:
[tex]\[ g(x) = \left(\frac{3}{4}\right)^2 \][/tex]
This is a continuous constant function.
For [tex]\( x \geq 0 \)[/tex]:
[tex]\[ g(x) = -x^2 \][/tex]
This is a continuous polynomial function.
However, we need to examine the point [tex]\( x = 0 \)[/tex] to check for continuity:
- From the left, as [tex]\( x \)[/tex] approaches 0, [tex]\( g(x) = \left(\frac{3}{4}\right)^2 \)[/tex].
- From the right, as [tex]\( x \)[/tex] approaches 0, [tex]\( g(0) = 0 \)[/tex].
The left and right limits at [tex]\( x = 0 \)[/tex] are not equal (since [tex]\( \left(\frac{3}{4}\right)^2 \neq 0 \)[/tex]). Thus, there is a discontinuity at [tex]\( x = 0 \)[/tex].
So, function [tex]\( g \)[/tex] is not continuous.
### Summary
Function [tex]\( g \)[/tex] has:
- 1 [tex]\( x \)[/tex]-intercept.
- 1 [tex]\( y \)[/tex]-intercept.
- The function is decreasing on its effective interval.
- The function is not continuous.
Thus, the correct answers to fill in the blanks are:
1. [tex]\( 1 \)[/tex] for [tex]\( x \)[/tex]-intercepts.
2. [tex]\( 1 \)[/tex] for [tex]\( y \)[/tex]-intercepts.
3. "is" for decreasing.
4. "not" for continuous.
