Inverse Functions And Equations Exploring F(x) And G(x)

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In the realm of mathematics, functions play a crucial role in describing relationships between variables. Understanding the properties of functions, including their inverses, is fundamental to solving various mathematical problems. This article delves into the intricacies of two specific functions, f(x) = (3x + 11) / (x - 3) and g(x) = (x - 3) / 2, exploring their inverses and investigating scenarios where a function equals the inverse of another.

Finding the Inverse of f(x)

The concept of an inverse function is central to our exploration. The inverse of a function, denoted as f-1(x), essentially reverses the operation of the original function. To find the inverse of f(x), we follow a systematic approach.

  1. Replace f(x) with y: This substitution helps simplify the equation and makes it easier to manipulate. So, we rewrite f(x) = (3x + 11) / (x - 3) as y = (3x + 11) / (x - 3).
  2. Swap x and y: This step is the core of finding the inverse. By interchanging x and y, we are essentially reversing the roles of the input and output. This gives us x = (3y + 11) / (y - 3).
  3. Solve for y: Our goal now is to isolate y on one side of the equation. This involves algebraic manipulation. Multiplying both sides by (y - 3), we get x(y - 3) = 3y + 11. Expanding the left side, we have xy - 3x = 3y + 11. Rearranging the terms to group y terms together, we get xy - 3y = 3x + 11. Factoring out y, we have y(x - 3) = 3x + 11. Finally, dividing both sides by (x - 3), we obtain y = (3x + 11) / (x - 3).
  4. Replace y with f-1(x): This final step expresses the inverse function in standard notation. Therefore, f-1(x) = (3x + 11) / (x - 3).

Interestingly, we observe that the inverse function f-1(x) is identical to the original function f(x). This implies that f(x) is its own inverse. Functions with this property are known as self-inverse functions or involutory functions. This unique characteristic has implications for the solutions we will find later.

Exploring the Inverse of g(x)

Now, let's determine the inverse of the function g(x) = (x - 3) / 2. We follow the same steps as before:

  1. Replace g(x) with y: y = (x - 3) / 2
  2. Swap x and y: x = (y - 3) / 2
  3. Solve for y: Multiplying both sides by 2, we get 2x = y - 3. Adding 3 to both sides, we isolate y: y = 2x + 3.
  4. Replace y with g-1(x): Thus, g-1(x) = 2x + 3.

Solving the Functional Equation f(x) = g-1(x)

The next part of our investigation involves finding the values of x for which f(x) = g-1(x). This means we need to solve the equation:

(3x + 11) / (x - 3) = 2x + 3

To solve this equation, we first multiply both sides by (x - 3) to eliminate the fraction:

3x + 11 = (2x + 3)(x - 3)

Expanding the right side, we get:

3x + 11 = 2x2 - 6x + 3x - 9

Simplifying and rearranging the terms, we obtain a quadratic equation:

2x2 - 6x - 20 = 0

Dividing the entire equation by 2, we further simplify it:

x2 - 3x - 10 = 0

Now, we can factor the quadratic equation:

(x - 5)(x + 2) = 0

This gives us two possible solutions for x:

  • x = 5
  • x = -2

Therefore, the values of x for which f(x) = g-1(x) are 5 and -2. We can verify these solutions by substituting them back into the original equation.

Exploring the Function h(x) = 5 - 6/x

Finally, let's consider the function h(x) = 5 - 6/x and determine its value for a specific input. This type of function, where a constant is divided by a variable, is known as a rational function. To find the value of h(x) for a specific input, we simply substitute the input value for x in the function's expression.

For instance, if we want to find h(2), we substitute x = 2 into the function:

h(2) = 5 - 6/2 = 5 - 3 = 2

Similarly, we can find h(3):

h(3) = 5 - 6/3 = 5 - 2 = 3

And h(-1):

h(-1) = 5 - 6/(-1) = 5 + 6 = 11

The value of the function h(x) depends on the input value of x. As x approaches infinity, the term 6/x approaches zero, and h(x) approaches 5. Conversely, as x approaches zero, the term 6/x becomes very large, and h(x) approaches either positive or negative infinity, depending on the sign of x.

Conclusion

This exploration has provided a comprehensive understanding of inverse functions and functional equations. We successfully determined the inverse of f(x) and found it to be a self-inverse function. We also calculated the inverse of g(x) and solved the equation f(x) = g-1(x), finding two solutions. Finally, we analyzed the function h(x) and determined its values for various inputs.

Understanding the concepts of inverse functions and functional equations is crucial for further studies in mathematics, particularly in calculus and analysis. The ability to manipulate functions, find their inverses, and solve equations involving functions is a fundamental skill for any aspiring mathematician or scientist. This article serves as a valuable resource for anyone seeking to deepen their understanding of these important mathematical concepts.

  • Functions
  • Inverse functions
  • Functional equations
  • f(x) = (3x + 11) / (x - 3)
  • g(x) = (x - 3) / 2
  • f-1(x)
  • g-1(x)
  • Self-inverse function
  • h(x) = 5 - 6/x
  • Rational function
  • Solving equations
  • Algebraic manipulation
  • Quadratic equation
  • Find the inverse of the function f(x) = (3x + 11) / (x - 3), where x ≠ 3.
  • Given f(x) = (3x + 11) / (x - 3) and g(x) = (x - 3) / 2, find f-1(x).
  • If f(x) = g-1(x), find the values of x, given f(x) = (3x + 11) / (x - 3) and g(x) = (x - 3) / 2.
  • If f(x) = 5 - 6/x, what is the value of f(x) for different values of x?