Suppose that the Functions g and h are Defined as Follows

Oct 13, 2025 by pinkandtrash.com 7 views

Suppose that the Functions g and h are Defined as Follows

Introduction

In this article, we will explore the concept of function composition and how to find the value of a composite function. We will be given two functions, g(x) and h(x), and we will need to find the value of the composite function (g/h)(x) at a specific point, x = -1.

Function Definitions

The two functions, g(x) and h(x), are defined as follows:

g(x) = -5 + 2x^2
h(x) = 2 - 5x

Part (a) - Finding the Value of (g/h)(-1)

To find the value of (g/h)(-1), we need to substitute x = -1 into the composite function (g/h)(x). However, before we can do that, we need to find the composite function (g/h)(x).

The composite function (g/h)(x) is defined as:

(g/h)(x) = g(x) / h(x)

Substituting the definitions of g(x) and h(x), we get:

(g/h)(x) = (-5 + 2x^2) / (2 - 5x)

Now that we have the composite function (g/h)(x), we can find the value of (g/h)(-1) by substituting x = -1 into the composite function:

(g/h)(-1) = (-5 + 2(-1)^2) / (2 - 5(-1))

Simplifying the expression, we get:

(g/h)(-1) = (-5 + 2) / (2 + 5)

(g/h)(-1) = -3 / 7

Therefore, the value of (g/h)(-1) is -3/7.

Part (b) - Finding All Values that are in the Domain of (g/h)

To find all values that are in the domain of (g/h), we need to find the values of x for which the denominator of the composite function (g/h)(x) is not equal to zero.

The denominator of the composite function (g/h)(x) is:

2 - 5x

To find the values of x for which the denominator is not equal to zero, we need to solve the equation:

2 - 5x ≠ 0

Solving the equation, we get:

x ≠ 2/5

Therefore, the values of x for which the denominator of the composite function (g/h)(x) is not equal to zero are all real numbers except x = 2/5.

Conclusion

In this article, we found the value of the composite function (g/h)(-1) and all values that are in the domain of (g/h). We used the definitions of the functions g(x) and h(x) to find the composite function (g/h)(x) and then substituted x = -1 into the composite function to find the value of (g/h)(-1). We also found the values of x for which the denominator of the composite function (g/h)(x) is not equal to zero.

References

[1] Calculus, James Stewart, 7th edition
[2] Algebra and Trigonometry, Michael Sullivan, 7th edition

Future Work

In the future, we can explore more complex function compositions and find the values of composite functions at specific points. We can also investigate the properties of composite functions, such as their derivatives and integrals.

Keywords

Function composition
Composite function
Domain of a function
Algebra
Calculus

External Links

[1] Khan Academy - Function Composition
[2] Mathway - Function Composition Calculator

License

This article is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.
Suppose that the Functions g and h are Defined as Follows: Q&A

Introduction

In this article, we will continue to explore the concept of function composition and answer some common questions related to the topic. We will be given two functions, g(x) and h(x), and we will need to find the value of the composite function (g/h)(x) at a specific point, x = -1.

Q&A

Q: What is function composition?

A: Function composition is the process of combining two or more functions to create a new function. The resulting function takes the input of one function and uses it as the input for the other function.

Q: How do you find the composite function (g/h)(x)?

A: To find the composite function (g/h)(x), you need to substitute the definition of g(x) into the definition of h(x). This will give you a new function that takes x as input and returns the result of g(x) divided by h(x).

Q: What is the domain of a composite function?

A: The domain of a composite function is the set of all input values for which the function is defined. In the case of the composite function (g/h)(x), the domain is all real numbers except x = 2/5.

Q: How do you find the value of a composite function at a specific point?

A: To find the value of a composite function at a specific point, you need to substitute the value of x into the composite function and simplify the expression.

Q: What is the difference between a function and a composite function?

A: A function is a relation between a set of inputs and a set of possible outputs. A composite function is a new function that is created by combining two or more existing functions.

Q: Can a composite function have multiple inputs?

A: Yes, a composite function can have multiple inputs. For example, if we have two functions f(x) and g(x), we can create a composite function (f ∘ g)(x) that takes x as input and returns the result of f(g(x)).

Q: How do you find the inverse of a composite function?

A: To find the inverse of a composite function, you need to swap the input and output values and then solve for the new input value.

Q: What is the importance of function composition in real-world applications?

A: Function composition is an important concept in many real-world applications, including physics, engineering, and computer science. It allows us to model complex systems and solve problems that involve multiple variables.

Conclusion

In this article, we have answered some common questions related to function composition and provided examples to illustrate the concept. We have also discussed the importance of function composition in real-world applications.