Welcome to Westonci.ca, your ultimate destination for finding answers to a wide range of questions from experts. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
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.
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.
