Solving Composite Functions Find X When (f ∘ G)(x) = -17
In this article, we will delve into the process of finding the values of x that satisfy a given set of conditions involving composite functions. Specifically, we will explore a scenario where we have two functions, f(x) and g(x), and a condition on their composite function (f ∘ g)(x). Our goal is to determine the set of all x values that make the given condition true. This exploration involves understanding function composition, algebraic manipulation, and problem-solving techniques within the realm of mathematics.
Understanding Composite Functions
Before we dive into the specific problem, let's briefly review the concept of composite functions. A composite function is formed when one function is applied to the result of another function. In other words, if we have two functions, f(x) and g(x), the composite function (f ∘ g)(x) is defined as f(g(x)). This means we first evaluate the function g(x), and then we use the result as the input for the function f(x). Composite functions are a fundamental concept in mathematics, allowing us to combine and analyze functions in more complex ways. They play a crucial role in various areas of mathematics, including calculus, analysis, and abstract algebra. Understanding composite functions is essential for solving a wide range of problems and gaining a deeper understanding of mathematical relationships.
Key Concepts of Composite Functions
- Order of Operations: In a composite function, the inner function is evaluated first, and its output becomes the input for the outer function.
- Domain and Range: The domain of a composite function (f ∘ g)(x) is the set of all x values in the domain of g(x) such that g(x) is in the domain of f(x). The range of the composite function is the set of all possible output values.
- Composition Notation: The notation (f ∘ g)(x) represents the composition of f and g, where g is applied first, followed by f.
Example
To illustrate the concept, let's consider two simple functions:
- f(x) = x²
- g(x) = x + 1
The composite function (f ∘ g)(x) would be:
(f ∘ g)(x) = f(g(x)) = f(x + 1) = (x + 1)²
This example demonstrates how the output of g(x), which is (x + 1), becomes the input for f(x).
Problem Statement
Now, let's focus on the specific problem we aim to solve. We are given two functions:
- f(x) = 3x - 8
- g(x) = 2x + 7
We are also given the condition that the composite function (f ∘ g)(x) is equal to -17:
(f ∘ g)(x) = -17
Our task is to find all values of x that satisfy this condition. This involves determining the set of solutions for the equation formed by the composite function and the given value. Solving this problem requires a step-by-step approach that combines the concepts of function composition and algebraic manipulation. By carefully applying these techniques, we can identify the x values that make the equation true and thus satisfy the given condition. The solution set will represent the values of x that fulfill the specified criteria.
Solution Approach
To find the values of x that satisfy the given conditions, we will follow a step-by-step approach:
- Find the composite function (f ∘ g)(x): This involves substituting g(x) into f(x).
- Set the composite function equal to -17: This translates the given condition into an equation.
- Solve the equation for x: This involves algebraic manipulation to isolate x and find its possible values.
- Express the solution set: This represents the set of all x values that satisfy the given conditions.
Step 1: Find the Composite Function (f ∘ g)(x)
To find the composite function (f ∘ g)(x), we substitute g(x) into f(x). Given:
- f(x) = 3x - 8
- g(x) = 2x + 7
We have:
(f ∘ g)(x) = f(g(x)) = f(2x + 7)
Now, we replace x in f(x) with (2x + 7):
f(2x + 7) = 3(2x + 7) - 8
Next, we simplify the expression:
3(2x + 7) - 8 = 6x + 21 - 8 = 6x + 13
Therefore, the composite function (f ∘ g)(x) is:
(f ∘ g)(x) = 6x + 13
This composite function represents the result of applying g(x) first and then applying f(x) to the output. It is a linear function of x, and we will use it to solve the equation in the next step.
Step 2: Set the Composite Function Equal to -17
Now that we have found the composite function (f ∘ g)(x) = 6x + 13 , we can set it equal to -17, as given in the problem statement:
(f ∘ g)(x) = -17
Substituting the expression for the composite function, we get:
6x + 13 = -17
This equation represents the condition that the composite function must satisfy. Our next step is to solve this equation for x, which will give us the values of x that make the composite function equal to -17. This equation is a linear equation in one variable, and we can solve it using basic algebraic techniques. The solution will provide us with the x value(s) that satisfy the given condition of the problem. By setting up this equation, we have translated the problem into a solvable form, allowing us to find the desired values of x.
Step 3: Solve the Equation for x
To solve the equation 6x + 13 = -17 for x, we will use algebraic manipulation to isolate x on one side of the equation. First, we subtract 13 from both sides of the equation:
6x + 13 - 13 = -17 - 13
This simplifies to:
6x = -30
Next, we divide both sides of the equation by 6:
6x / 6 = -30 / 6
This gives us:
x = -5
Therefore, the solution to the equation 6x + 13 = -17 is x = -5. This means that when x is equal to -5, the composite function (f ∘ g)(x) will be equal to -17. This value of x satisfies the given condition of the problem. The algebraic steps we followed ensured that we isolated x correctly, leading us to the accurate solution. The value x = -5 is a critical finding, as it represents the input that makes the composite function behave as specified in the problem statement.
Step 4: Express the Solution Set
Now that we have found the value of x that satisfies the given conditions, we can express the solution set. The solution set is the set of all x values that make the equation (f ∘ g)(x) = -17 true. In this case, we found only one value of x that satisfies the equation, which is x = -5. Therefore, the solution set is a singleton set containing only the value -5. We can express the solution set using set notation as follows:
{ -5 }
This set notation clearly indicates that the solution set consists of the single element -5. This completes the problem-solving process, as we have successfully identified the value of x that satisfies the given conditions and expressed it in the appropriate set notation. The solution set provides a concise and accurate representation of the answer to the problem, indicating all values of x that make the composite function equal to -17.
Final Answer
Therefore, the solution set for the problem is:
{ -5 }
This indicates that the only value of x that satisfies the given conditions is x = -5. This answer is the culmination of the step-by-step solution process, where we first found the composite function, then set it equal to the given value, solved the resulting equation, and finally expressed the solution set. The final answer provides a clear and concise solution to the problem, confirming the value of x that makes the composite function equal to -17. The solution set is a fundamental concept in mathematics, and understanding how to find and express it is crucial for solving various types of problems involving equations and functions.
Conclusion
In this article, we have successfully found all values of x that satisfy the given conditions involving composite functions. We started by understanding the concept of composite functions and their properties. Then, we applied a systematic approach to solve the problem, which involved finding the composite function, setting it equal to the given value, solving the resulting equation, and expressing the solution set. The final answer, { -5 }, represents the set of all x values that make the composite function (f ∘ g)(x) equal to -17. This problem-solving process demonstrates the importance of understanding function composition and algebraic manipulation in mathematics. By applying these techniques, we can solve a variety of problems involving functions and equations. The ability to work with composite functions is a valuable skill in mathematics, and this article has provided a detailed example of how to approach and solve such problems effectively.
