Answer :
Sure! Let's start by understanding the basic function and the transformations applied to it to obtain the final function \( g(x) = (x + 2)^3 + 6 \).
### Step-by-step Transformation:
1. Basic Function
- Start with the basic cubic function \( f(x) = x^3 \). This function has a typical "S" shaped curve with domain \((- \infty, \infty)\) and range \((- \infty, \infty)\).
2. Horizontal Shift
- Next, consider the transformation \( f(x) = (x + 2)^3 \). This represents a horizontal shift. Specifically, \( x \) is replaced by \((x + 2)\), which shifts the graph 2 units to the left.
- So, the graph of \( (x + 2)^3 \) will be the same "S" shaped curve as \( x^3 \), but moved 2 units to the left.
3. Vertical Shift
- Now, we apply the transformation \( g(x) = (x + 2)^3 + 6 \). This means we take the function \( (x + 2)^3 \) and shift it vertically up by 6 units.
- So, every point on the graph of \( (x + 2)^3 \) is moved up by 6 units.
### Graphing the Function:
- To graph \( g(x) = (x + 2)^3 + 6 \):
1. Start with the graph of \( y = x^3 \).
2. Shift the entire graph 2 units to the left to get \( y = (x + 2)^3 \).
3. Move the resultant graph 6 units up to get the final graph of \( g(x) = (x + 2)^3 + 6 \).
### Domain and Range:
- Domain:
- The cubic function \( x^3 \) has an unrestricted domain \((- \infty, \infty)\).
- Shifting it horizontally and vertically does not change the fact that \( x \) can take any real value.
- Therefore, the domain of \( g(x) = (x + 2)^3 + 6 \) is \((- \infty, \infty)\).
- Range:
- Similarly, the basic cubic function \( x^3 \) spans all real numbers in the range \((- \infty, \infty)\).
- The vertical shift moves the graph up and down but does not restrict the range.
- Therefore, the range of \( g(x) = (x + 2)^3 + 6 \) is \((- \infty, \infty)\).
So, both the domain and range of the function \( g(x) = (x+2)^3 + 6 \) are \((- \infty, \infty)\).
In conclusion:
- The Domain of \( g(x) = (x+2)^3 + 6 \) is \( (-\infty, \infty) \).
- The Range of \( g(x) = (x+2)^3 + 6 \) is \( (-\infty, \infty) \).
You can use graphing tools to visualize this step-by-step transformation.
### Step-by-step Transformation:
1. Basic Function
- Start with the basic cubic function \( f(x) = x^3 \). This function has a typical "S" shaped curve with domain \((- \infty, \infty)\) and range \((- \infty, \infty)\).
2. Horizontal Shift
- Next, consider the transformation \( f(x) = (x + 2)^3 \). This represents a horizontal shift. Specifically, \( x \) is replaced by \((x + 2)\), which shifts the graph 2 units to the left.
- So, the graph of \( (x + 2)^3 \) will be the same "S" shaped curve as \( x^3 \), but moved 2 units to the left.
3. Vertical Shift
- Now, we apply the transformation \( g(x) = (x + 2)^3 + 6 \). This means we take the function \( (x + 2)^3 \) and shift it vertically up by 6 units.
- So, every point on the graph of \( (x + 2)^3 \) is moved up by 6 units.
### Graphing the Function:
- To graph \( g(x) = (x + 2)^3 + 6 \):
1. Start with the graph of \( y = x^3 \).
2. Shift the entire graph 2 units to the left to get \( y = (x + 2)^3 \).
3. Move the resultant graph 6 units up to get the final graph of \( g(x) = (x + 2)^3 + 6 \).
### Domain and Range:
- Domain:
- The cubic function \( x^3 \) has an unrestricted domain \((- \infty, \infty)\).
- Shifting it horizontally and vertically does not change the fact that \( x \) can take any real value.
- Therefore, the domain of \( g(x) = (x + 2)^3 + 6 \) is \((- \infty, \infty)\).
- Range:
- Similarly, the basic cubic function \( x^3 \) spans all real numbers in the range \((- \infty, \infty)\).
- The vertical shift moves the graph up and down but does not restrict the range.
- Therefore, the range of \( g(x) = (x + 2)^3 + 6 \) is \((- \infty, \infty)\).
So, both the domain and range of the function \( g(x) = (x+2)^3 + 6 \) are \((- \infty, \infty)\).
In conclusion:
- The Domain of \( g(x) = (x+2)^3 + 6 \) is \( (-\infty, \infty) \).
- The Range of \( g(x) = (x+2)^3 + 6 \) is \( (-\infty, \infty) \).
You can use graphing tools to visualize this step-by-step transformation.
