Representation of Functions

Oct 13, 2025 by magicalceylontours.com 4 views

Introduction

In mathematics, functions are used to describe the relationship between variables. When dealing with functions, it's essential to understand their properties and behavior. In this article, we will explore the given representations of $$f(x), g(x),$$ and $$h(x)$$ and help you select the correct answer from each drop-down menu.

Representation of Functions

Function f(x)

The first function is represented as $$f(x) = 4 \sqrt[3]{x-3} + 5$$. This function has a cube root term, which indicates that it is a radical function. The cube root term is $$\sqrt[3]{x-3}$$, and it is multiplied by 4 and then added to 5.

Function g(x)

The second function is a cube root function with a scale of 2, represented as $$g(x) = 2\sqrt[3]{x-2}$$. This function also has a cube root term, but it is scaled by a factor of 2.

Function h(x)

The third function is not explicitly given, but we can infer its properties based on the given functions.

Properties of Functions

Domain and Range

The domain of a function is the set of all possible input values, while the range is the set of all possible output values. For the given functions, we can determine their domains and ranges based on the properties of the cube root function.

For $$f(x) = 4 \sqrt[3]{x-3} + 5$$, the domain is all real numbers greater than or equal to 3, since the cube root term is defined only for non-negative values. The range is all real numbers greater than or equal to 5.
For $$g(x) = 2\sqrt[3]{x-2}$$, the domain is all real numbers greater than or equal to 2, since the cube root term is defined only for non-negative values. The range is all real numbers greater than or equal to 0.

Scale and Shift

The scale of a function refers to the factor by which the input values are multiplied, while the shift refers to the horizontal or vertical translation of the function.

For $$f(x) = 4 \sqrt[3]{x-3} + 5$$, the scale is 4, and the shift is 3 units to the right and 5 units up.
For $$g(x) = 2\sqrt[3]{x-2}$$, the scale is 2, and the shift is 2 units to the right.

Composition of Functions

The composition of functions refers to the process of combining two or more functions to create a new function. For the given functions, we can determine their compositions based on the properties of the cube root function.

For $$f(x) = 4 \sqrt[3]{x-3} + 5$$ and $$g(x) = 2\sqrt[3]{x-2}$$, the composition $$f(g(x))$$ would result in a new function with a scale of 8 and a shift of 3 units to the right and 5 units up.

Selecting the Correct Answer from Each Drop-Down Menu

Based on the properties of the given functions, we can select the correct answer from each drop-down menu.

Drop-Down Menu 1

What is the domain of $$f(x) = 4 \sqrt[3]{x-3} + 5$$?
- All real numbers greater than or equal to 3
- All real numbers less than or equal to 3
- All real numbers greater than or equal to 5
- All real numbers less than or equal to 5

Correct answer: All real numbers greater than or equal to 3

Drop-Down Menu 2

What is the range of $$g(x) = 2\sqrt[3]{x-2}$$?
- All real numbers greater than or equal to 0
- All real numbers less than or equal to 0
- All real numbers greater than or equal to 2
- All real numbers less than or equal to 2

Correct answer: All real numbers greater than or equal to 0

Drop-Down Menu 3

What is the scale of $$f(x) = 4 \sqrt[3]{x-3} + 5$$?
- 2
- 4
- 8
- 16

Correct answer: 4

Drop-Down Menu 4

What is the shift of $$g(x) = 2\sqrt[3]{x-2}$$?
- 2 units to the left
- 2 units to the right
- 3 units to the left
- 3 units to the right

Correct answer: 2 units to the right

Conclusion

In conclusion, understanding the properties of functions is essential in mathematics. By analyzing the given representations of $$f(x), g(x),$$ and $$h(x)$$, we can determine their domains, ranges, scales, and shifts. By selecting the correct answer from each drop-down menu, we can demonstrate our understanding of these properties and their applications in mathematics.

References

[1] "Functions" by Khan Academy
[2] "Radical Functions" by Math Open Reference
[3] "Composition of Functions" by Purplemath
Q&A: Understanding and Selecting the Correct Answer from Each Drop-Down Menu ====================================================================

Q: What is the domain of the function f(x) = 4 \sqrt[3]{x-3} + 5?

A: The domain of the function f(x) = 4 \sqrt[3]{x-3} + 5 is all real numbers greater than or equal to 3. This is because the cube root term \sqrt[3]{x-3} is defined only for non-negative values, and x-3 must be greater than or equal to 0 for the cube root to be defined.

Q: What is the range of the function g(x) = 2\sqrt[3]{x-2}?

A: The range of the function g(x) = 2\sqrt[3]{x-2} is all real numbers greater than or equal to 0. This is because the cube root term \sqrt[3]{x-2} is always non-negative, and multiplying it by 2 will result in a non-negative value.

Q: What is the scale of the function f(x) = 4 \sqrt[3]{x-3} + 5?

A: The scale of the function f(x) = 4 \sqrt[3]{x-3} + 5 is 4. This is because the cube root term \sqrt[3]{x-3} is multiplied by 4, resulting in a scale factor of 4.

Q: What is the shift of the function g(x) = 2\sqrt[3]{x-2}?

A: The shift of the function g(x) = 2\sqrt[3]{x-2} is 2 units to the right. This is because the cube root term \sqrt[3]{x-2} is shifted 2 units to the right, resulting in a shift of 2 units to the right.

Q: What is the composition of the functions f(x) = 4 \sqrt[3]{x-3} + 5 and g(x) = 2\sqrt[3]{x-2}?

A: The composition of the functions f(x) = 4 \sqrt[3]{x-3} + 5 and g(x) = 2\sqrt[3]{x-2} is f(g(x)) = 8\sqrt[3]{x-2} + 5. This is because the function g(x) = 2\sqrt[3]{x-2} is composed with the function f(x) = 4 \sqrt[3]{x-3} + 5, resulting in a new function with a scale factor of 8 and a shift of 5 units up.

Q: What is the domain of the composition f(g(x))?

A: The domain of the composition f(g(x)) = 8\sqrt[3]{x-2} + 5 is all real numbers greater than or equal to 2. This is because the cube root term \sqrt[3]{x-2} is defined only for non-negative values, and x-2 must be greater than or equal to 0 for the cube root to be defined.

Q: What is the range of the composition f(g(x))?

A: The range of the composition f(g(x)) = 8\sqrt[3]{x-2} + 5 is all real numbers greater than or equal to 5. This is because the cube root term \sqrt[3]{x-2} is always non-negative, and multiplying it by 8 and adding 5 will result in a non-negative value.

Q: What is the scale of the composition f(g(x))?

A: The scale of the composition f(g(x)) = 8\sqrt[3]{x-2} + 5 is 8. This is because the cube root term \sqrt[3]{x-2} is multiplied by 8, resulting in a scale factor of 8.

Q: What is the shift of the composition f(g(x))?

A: The shift of the composition f(g(x)) = 8\sqrt[3]{x-2} + 5 is 2 units to the right and 5 units up. This is because the cube root term \sqrt[3]{x-2} is shifted 2 units to the right, and the function f(x) = 4 \sqrt[3]{x-3} + 5 is shifted 5 units up.

Conclusion

In conclusion, understanding the properties of functions is essential in mathematics. By analyzing the given representations of f(x), g(x), and h(x), we can determine their domains, ranges, scales, and shifts. By selecting the correct answer from each drop-down menu, we can demonstrate our understanding of these properties and their applications in mathematics.