Given The Functions:${ F(x) = X^3 - 2x^2 + 8 }$ { G(x) = 7x - 3 \} ${ H(x) = X^2 - 4x }$Find { (f - G + H)(x)$}$.

Introduction

In mathematics, functions are used to describe relationships between variables. When we have multiple functions, we can combine them to create new functions. In this article, we will explore how to find the composition of functions, specifically the function $(f - g + h)(x)$, given the functions $f(x) = x^3 - 2x^2 + 8$, $g(x) = 7x - 3$, and $h(x) = x^2 - 4x$.

Understanding the Functions

Before we can find the composition of functions, we need to understand the individual functions. The function $f(x) = x^3 - 2x^2 + 8$ is a cubic function, which means it has a degree of 3. The function $g(x) = 7x - 3$ is a linear function, which means it has a degree of 1. The function $h(x) = x^2 - 4x$ is a quadratic function, which means it has a degree of 2.

Finding the Composition of Functions

To find the composition of functions, we need to substitute the expression for one function into another function. In this case, we need to find $(f - g + h)(x)$. To do this, we will first find the expression for $f(x) - g(x)$, then add $h(x)$ to the result.

Step 1: Find the Expression for $f(x) - g(x)$

To find the expression for $f(x) - g(x)$, we need to subtract the expression for $g(x)$ from the expression for $f(x)$. This means we will subtract $7x - 3$ from $x^3 - 2x^2 + 8$.

import sympy as sp
x = sp.symbols('x')

f = x3 - 2*x2 + 8
g = 7*x - 3

expr = f - g
print(expr)

Step 2: Simplify the Expression

After finding the expression for $f(x) - g(x)$, we need to simplify it. This means we need to combine like terms and eliminate any unnecessary terms.

# Simplify the expression
simplified_expr = sp.simplify(expr)
print(simplified_expr)

Step 3: Add $h(x)$ to the Result

Now that we have the simplified expression for $f(x) - g(x)$, we can add $h(x)$ to the result. This means we will add $x^2 - 4x$ to the simplified expression.

# Define the function h(x)
h = x**2 - 4*x

final_expr = simplified_expr + h
print(final_expr)

Conclusion

In this article, we explored how to find the composition of functions, specifically the function $(f - g + h)(x)$, given the functions $f(x) = x^3 - 2x^2 + 8$, $g(x) = 7x - 3$, and $h(x) = x^2 - 4x$. We used Python code to find the expression for $f(x) - g(x)$, simplify the expression, and add $h(x)$ to the result. The final expression for $(f - g + h)(x)$ is $x^3 - 2x^2 + 8 - 7x + 3 + x^2 - 4x$, which simplifies to $x^3 - 3x^2 - 11x + 11$.

Final Answer

The final answer is $\boxed{x^3 - 3x^2 - 11x + 11}$.

Introduction

In our previous article, we explored how to find the composition of functions, specifically the function $(f - g + h)(x)$, given the functions $f(x) = x^3 - 2x^2 + 8$, $g(x) = 7x - 3$, and $h(x) = x^2 - 4x$. In this article, we will answer some frequently asked questions about finding the composition of functions.

Q: What is the composition of functions?

A: The composition of functions is the process of combining two or more functions to create a new function. This is done by substituting the expression for one function into another function.

Q: How do I find the composition of functions?

A: To find the composition of functions, you need to follow these steps:

1. Define the individual functions.
2. Substitute the expression for one function into another function.
3. Simplify the resulting expression.
4. Add or subtract the resulting expression from other functions as needed.

Q: What is the difference between function composition and function addition?

A: Function composition is the process of combining two or more functions to create a new function, while function addition is the process of adding two or more functions together. For example, if we have two functions $f(x) = x^2 + 1$ and $g(x) = x + 2$, the composition of functions would be $(f \circ g)(x) = f(g(x)) = (x + 2)^2 + 1$, while the addition of functions would be $f(x) + g(x) = x^2 + 1 + x + 2$.

Q: Can I use function composition to find the inverse of a function?

A: Yes, you can use function composition to find the inverse of a function. To do this, you need to swap the input and output variables of the function and then simplify the resulting expression.

Q: How do I use function composition to solve equations?

A: To use function composition to solve equations, you need to substitute the expression for one function into another function and then solve for the variable. For example, if we have the equation $f(x) = g(x)$, we can substitute the expression for $f(x)$ into the equation and then solve for $x$.

Q: Can I use function composition to find the derivative of a function?

A: Yes, you can use function composition to find the derivative of a function. To do this, you need to use the chain rule of differentiation, which states that if we have a composite function $f(g(x))$, the derivative of the function is given by $f'(g(x)) \cdot g'(x)$.

Q: How do I use function composition to find the integral of a function?

A: To use function composition to find the integral of a function, you need to use the fundamental theorem of calculus, which states that the integral of a function is equal to the antiderivative of the function.

Conclusion

In this article, we answered some frequently asked questions about finding the composition of functions. We hope that this article has been helpful in understanding the concept of function composition and how to use it to solve equations and find the derivative and integral of functions.

Final Answer

The final answer is that function composition is a powerful tool that can be used to solve equations, find the derivative and integral of functions, and more. By understanding how to use function composition, you can become a more proficient mathematician and problem-solver.