Consider The Functions F ( X ) = − 2 X − 3 F(x)=-2x-3 F ( X ) = − 2 X − 3 And G ( X ) = X − 2 G(x)=\sqrt{x-2} G ( X ) = X − 2 ​ . Determine Each Of The Following:1. F ∘ G ( X ) = − 2 X − 2 − 3 F \circ G(x) = -2 \sqrt{x-2} - 3 F ∘ G ( X ) = − 2 X − 2 ​ − 3 - Give The Domain Of F ∘ G ( X F \circ G(x F ∘ G ( X ]. [ ] 2. G ∘ F ( X ) = − 2 X − 5 G \circ F(x) = \sqrt{-2x-5} G ∘ F ( X ) = − 2 X − 5 ​

Introduction

In mathematics, the composition of functions is a fundamental concept that allows us to combine two or more functions to create a new function. Given two functions, $f(x)$ and $g(x)$, the composition of $f$ and $g$ is denoted by $f \circ g(x)$ and is defined as $(f \circ g)(x) = f(g(x))$. In this article, we will explore the composition of two given functions, $f(x)=-2x-3$ and $g(x)=\sqrt{x-2}$, and determine the domain of the resulting composite function.

Composition of Functions

$f \circ g(x)$

To find the composition of $f$ and $g$, we need to substitute $g(x)$ into $f(x)$.

$f \circ g(x) = f(g(x)) = -2 \sqrt{x-2} - 3$

Domain of $f \circ g(x)$

To determine the domain of $f \circ g(x)$, we need to consider the restrictions imposed by both functions. The function $g(x) = \sqrt{x-2}$ requires that $x-2 \geq 0$, which implies that $x \geq 2$. Additionally, the function $f(x) = -2x-3$ has no restrictions on its domain.

However, since $g(x)$ is the inner function, its output must be within the domain of $f(x)$. Therefore, we need to ensure that $g(x) \geq -3$, which implies that $\sqrt{x-2} \geq -3$. Since the square root function is always non-negative, this inequality is always true.

Therefore, the domain of $f \circ g(x)$ is the same as the domain of $g(x)$, which is $x \geq 2$.

$g \circ f(x)$

To find the composition of $g$ and $f$, we need to substitute $f(x)$ into $g(x)$.

$g \circ f(x) = g(f(x)) = \sqrt{-2x-5}$

Domain of $g \circ f(x)$

To determine the domain of $g \circ f(x)$, we need to consider the restrictions imposed by both functions. The function $f(x) = -2x-3$ has no restrictions on its domain. However, the function $g(x) = \sqrt{x-2}$ requires that $x-2 \geq 0$, which implies that $x \geq 2$.

Since $f(x)$ is the inner function, its output must be within the domain of $g(x)$. Therefore, we need to ensure that $f(x) \geq 2$, which implies that $-2x-3 \geq 2$. Solving this inequality, we get $x \leq -\frac{5}{2}$.

Therefore, the domain of $g \circ f(x)$ is $x \leq -\frac{5}{2}$.

Conclusion

In conclusion, we have determined the composition of two given functions, $f(x)=-2x-3$ and $g(x)=\sqrt{x-2}$, and found the domain of the resulting composite function. We have also explored the composition of the functions in the reverse order, $g \circ f(x)$, and determined its domain.

References

• [1] "Composition of Functions" by Math Open Reference
• [2] "Domain of a Function" by Khan Academy

Further Reading

• [1] "Functions" by Wolfram MathWorld
• [2] "Composition of Functions" by Purplemath

Mathematical Notations

• $\circ$: Composition of functions
• $f(x)$: Function $f$ evaluated at $x$
• $g(x)$: Function $g$ evaluated at $x$
• $f \circ g(x)$: Composition of $f$ and $g$ evaluated at $x$
• $g \circ f(x)$: Composition of $g$ and $f$ evaluated at $x$
• $\sqrt{x-2}$: Square root function evaluated at $x-2$
• $x \geq 2$: Inequality $x$ is greater than or equal to $2$
• $x \leq -\frac{5}{2}$: Inequality $x$ is less than or equal to $-\frac{5}{2}$
  Composition of Functions: A Q&A Guide =====================================

Introduction

In our previous article, we explored the composition of two given functions, $f(x)=-2x-3$ and $g(x)=\sqrt{x-2}$, and determined the domain of the resulting composite function. In this article, we will answer some frequently asked questions about the composition of functions.

Q&A

Q: What is the composition of functions?

A: The composition of functions is a way of combining two or more functions to create a new function. Given two functions, $f(x)$ and $g(x)$, the composition of $f$ and $g$ is denoted by $f \circ g(x)$ and is defined as $(f \circ g)(x) = f(g(x))$.

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

A: To find the composition of two functions, you need to substitute the inner function into the outer function. For example, to find the composition of $f(x)=-2x-3$ and $g(x)=\sqrt{x-2}$, you would substitute $g(x)$ into $f(x)$ to get $f \circ g(x) = -2 \sqrt{x-2} - 3$.

Q: What is the domain of a composite function?

A: The domain of a composite function is the set of all possible input values for the inner function. In other words, the domain of the composite function is the same as the domain of the inner function.

Q: How do I determine the domain of a composite function?

A: To determine the domain of a composite function, you need to consider the restrictions imposed by both functions. You need to ensure that the output of the inner function is within the domain of the outer function.

Q: What is the difference between $f \circ g(x)$ and $g \circ f(x)$?

A: The composition of functions is not commutative, meaning that the order of the functions matters. $f \circ g(x)$ is different from $g \circ f(x)$, and the resulting composite functions will also be different.

Q: Can I compose more than two functions?

A: Yes, you can compose more than two functions. For example, you can compose three functions $f(x)$, $g(x)$, and $h(x)$ to get $(f \circ g \circ h)(x) = f(g(h(x)))$.

Q: What are some common applications of the composition of functions?

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

• Modeling real-world phenomena, such as population growth or chemical reactions
• Solving systems of equations
• Finding the inverse of a function
• Analyzing the behavior of complex systems

Conclusion

In conclusion, the composition of functions is a powerful tool for combining two or more functions to create a new function. By understanding the composition of functions, you can solve a wide range of problems in mathematics, science, and engineering.

References

• [1] "Composition of Functions" by Math Open Reference
• [2] "Domain of a Function" by Khan Academy
• [3] "Functions" by Wolfram MathWorld
• [4] "Composition of Functions" by Purplemath

Further Reading

• [1] "Functions" by MIT OpenCourseWare
• [2] "Composition of Functions" by University of California, Berkeley
• [3] "Mathematical Modeling" by University of Michigan

Mathematical Notations

• $\circ$: Composition of functions
• $f(x)$: Function $f$ evaluated at $x$
• $g(x)$: Function $g$ evaluated at $x$
• $f \circ g(x)$: Composition of $f$ and $g$ evaluated at $x$
• $g \circ f(x)$: Composition of $g$ and $f$ evaluated at $x$
• $\sqrt{x-2}$: Square root function evaluated at $x-2$
• $x \geq 2$: Inequality $x$ is greater than or equal to $2$
• $x \leq -\frac{5}{2}$: Inequality $x$ is less than or equal to $-\frac{5}{2}$