Find { (f \circ G)(0)$} . . . { \begin{array}{c} f(x) = 3x + 5 \\ g(x) = X^2 \\ (f \circ G)(0) = \end{array} \}

Introduction

In mathematics, the composition of two functions is a fundamental concept that plays a crucial role in various areas of study, including calculus, algebra, and analysis. The composition of two functions, denoted as $(f \circ g)(x)$, is a new function that is obtained by combining the two original functions. In this article, we will focus on finding the composition of two given functions, $f(x) = 3x + 5$ and $g(x) = x^2$, and then evaluate the resulting function at $x = 0$.

Understanding Function Composition

Before we dive into the problem, let's take a moment to understand the concept of function composition. Given two functions, $f(x)$ and $g(x)$, the composition of these functions is defined as:

$$(f \circ g)(x) = f(g(x))$$

This means that we first apply the function $g(x)$ to the input $x$, and then apply the function $f(x)$ to the result. In other words, we plug the output of $g(x)$ into $f(x)$.

Evaluating the Composition of Two Functions

Now that we have a good understanding of function composition, let's evaluate the composition of the two given functions, $f(x) = 3x + 5$ and $g(x) = x^2$. To do this, we need to substitute $g(x)$ into $f(x)$:

$$(f \circ g)(x) = f(g(x)) = 3(g(x)) + 5$$

Since $g(x) = x^2$, we can substitute this expression into the equation above:

$$(f \circ g)(x) = 3(x^2) + 5$$

Simplifying the expression, we get:

$$(f \circ g)(x) = 3x^2 + 5$$

Evaluating the Composition at $x = 0$

Now that we have the composition of the two functions, let's evaluate the resulting function at $x = 0$. To do this, we simply substitute $x = 0$ into the equation:

$$(f \circ g)(0) = 3(0^2) + 5$$

Simplifying the expression, we get:

$$(f \circ g)(0) = 3(0) + 5 = 0 + 5 = 5$$

Therefore, the value of the composition of the two functions at $x = 0$ is $5$.

Conclusion

In this article, we have seen how to find the composition of two functions and evaluate the resulting function at a given point. We have used the concept of function composition to combine two given functions, $f(x) = 3x + 5$ and $g(x) = x^2$, and then evaluated the resulting function at $x = 0$. The value of the composition of the two functions at $x = 0$ is $5$. We hope that this article has provided a clear and concise explanation of the concept of function composition and how to evaluate the resulting function at a given point.

Additional Examples

To further illustrate the concept of function composition, let's consider a few more examples.

Example 1

Find the composition of the two functions, $f(x) = 2x - 1$ and $g(x) = x^3$. Evaluate the resulting function at $x = 1$.

Solution

To find the composition of the two functions, we need to substitute $g(x)$ into $f(x)$:

$$(f \circ g)(x) = f(g(x)) = 2(g(x)) - 1$$

Since $g(x) = x^3$, we can substitute this expression into the equation above:

$$(f \circ g)(x) = 2(x^3) - 1$$

Simplifying the expression, we get:

$$(f \circ g)(x) = 2x^3 - 1$$

To evaluate the resulting function at $x = 1$, we simply substitute $x = 1$ into the equation:

$$(f \circ g)(1) = 2(1^3) - 1 = 2(1) - 1 = 2 - 1 = 1$$

Therefore, the value of the composition of the two functions at $x = 1$ is $1$.

Example 2

Find the composition of the two functions, $f(x) = x^2 + 1$ and $g(x) = 2x - 3$. Evaluate the resulting function at $x = 2$.

Solution

To find the composition of the two functions, we need to substitute $g(x)$ into $f(x)$:

$$(f \circ g)(x) = f(g(x)) = (g(x))^2 + 1$$

Since $g(x) = 2x - 3$, we can substitute this expression into the equation above:

$$(f \circ g)(x) = (2x - 3)^2 + 1$$

Simplifying the expression, we get:

$$(f \circ g)(x) = 4x^2 - 12x + 9 + 1 = 4x^2 - 12x + 10$$

To evaluate the resulting function at $x = 2$, we simply substitute $x = 2$ into the equation:

$$(f \circ g)(2) = 4(2)^2 - 12(2) + 10 = 4(4) - 24 + 10 = 16 - 24 + 10 = 2$$

Therefore, the value of the composition of the two functions at $x = 2$ is $2$.

Final Thoughts

Q: What is function composition?

A: Function composition is a way of combining two or more functions to create a new function. It involves applying one function to the output of another function.

Q: How do I find the composition of two functions?

A: To find the composition of two functions, you need to substitute the second function into the first function. For example, if we have two functions, f(x) and g(x), the composition of these functions is denoted as (f ∘ g)(x) = f(g(x)).

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

A: Function composition involves combining two or more functions to create a new function, while function evaluation involves finding the value of a function at a given point.

Q: Can I have multiple functions in a composition?

A: Yes, you can have multiple functions in a composition. For example, if we have three functions, f(x), g(x), and h(x), the composition of these functions is denoted as (f ∘ g ∘ h)(x) = f(g(h(x))).

Q: How do I evaluate the composition of two functions at a given point?

A: To evaluate the composition of two functions at a given point, you need to substitute the given point into the composition of the two functions. For example, if we have the composition (f ∘ g)(x) = f(g(x)) and we want to evaluate it at x = 2, we need to substitute x = 2 into the equation.

Q: What are some common applications of function composition?

A: Function composition has many applications in mathematics, science, and engineering. Some common applications include:

• Calculus: Function composition is used to find the derivative of a function.
• Algebra: Function composition is used to solve systems of equations.
• Analysis: Function composition is used to study the properties of functions.
• Computer Science: Function composition is used in programming languages to create new functions from existing ones.

Q: Can I use function composition with different types of functions?

A: Yes, you can use function composition with different types of functions, including:

• Linear functions: Function composition can be used with linear functions to create new linear functions.
• Polynomial functions: Function composition can be used with polynomial functions to create new polynomial functions.
• Rational functions: Function composition can be used with rational functions to create new rational functions.
• Trigonometric functions: Function composition can be used with trigonometric functions to create new trigonometric functions.

Q: How do I know if a function is composite or not?

A: A function is composite if it can be expressed as the composition of two or more other functions. To determine if a function is composite or not, you need to check if it can be written in the form (f ∘ g)(x) = f(g(x)).

Q: Can I have a function that is both composite and non-composite?

A: No, a function cannot be both composite and non-composite at the same time. If a function is composite, it means that it can be expressed as the composition of two or more other functions. If a function is non-composite, it means that it cannot be expressed as the composition of two or more other functions.

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

A: To find the inverse of a composite function, you need to find the inverse of each function in the composition separately. For example, if we have the composition (f ∘ g)(x) = f(g(x)), the inverse of this function is (g ∘ f)(x) = g(f(x)).

Q: Can I have a composite function that is one-to-one?

A: Yes, a composite function can be one-to-one. If a function is one-to-one, it means that each output value corresponds to exactly one input value. A composite function can be one-to-one if each function in the composition is one-to-one.

Q: How do I know if a composite function is one-to-one or not?

A: To determine if a composite function is one-to-one or not, you need to check if each function in the composition is one-to-one. If each function in the composition is one-to-one, then the composite function is also one-to-one.

Q: Can I have a composite function that is onto?

A: Yes, a composite function can be onto. If a function is onto, it means that each output value corresponds to at least one input value. A composite function can be onto if each function in the composition is onto.

Q: How do I know if a composite function is onto or not?

A: To determine if a composite function is onto or not, you need to check if each function in the composition is onto. If each function in the composition is onto, then the composite function is also onto.

Q: Can I have a composite function that is both one-to-one and onto?

A: Yes, a composite function can be both one-to-one and onto. If a function is both one-to-one and onto, it means that each output value corresponds to exactly one input value and each output value corresponds to at least one input value. A composite function can be both one-to-one and onto if each function in the composition is both one-to-one and onto.