Answered

At Westonci.ca, we connect you with experts who provide detailed answers to your most pressing questions. Start exploring now! Discover comprehensive solutions to your questions from a wide network of experts on our user-friendly platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.

Starting with the graph of a basic function, graph the following function using the techniques of shifting, compressing, stretching, and/or reflecting. Find the domain and range of the function.

[tex]\[ g(x) = (x + 2)^3 + 6 \][/tex]

Use the graphing tool to graph the equation.

Sagot :

GinnyAnswer
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.
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.