Finding The Inverse Of G(x) And Its Domain & Range

by Alex Johnson 51 views

Let's dive into the world of functions and explore how to find the inverse of a given function, as well as determine its domain and range. In this article, we'll tackle a specific example step-by-step, making the process clear and easy to understand. Our focus will be on the function g(x) = (7x - 4) / (3x + 8), a one-to-one function, which is crucial for having an inverse. We will find the inverse function g-1(x) and then specify its domain and range using interval notation. So, let's get started!

(a) Finding the Inverse Function g-1(x)

To find the inverse of a function, we essentially reverse the roles of x and y. Here’s the breakdown of the process for our function g(x) = (7x - 4) / (3x + 8):

  1. Replace g(x) with y: This makes the equation easier to manipulate. So, we have y = (7x - 4) / (3x + 8).
  2. Swap x and y: This is the key step in finding the inverse. Our equation becomes x = (7y - 4) / (3y + 8).
  3. Solve for y: Now, we need to isolate y on one side of the equation. This will give us the inverse function.
    • Multiply both sides by (3y + 8): This gets rid of the fraction, giving us x(3y + 8) = 7y - 4.
    • Distribute x: We have 3xy + 8x = 7y - 4.
    • Gather terms with y on one side and other terms on the other side: This gives us 3xy - 7y = -8x - 4.
    • Factor out y: We get y(3x - 7) = -8x - 4.
    • Divide by (3x - 7) to isolate y: This results in y = (-8x - 4) / (3x - 7).
  4. Replace y with g-1(x): This is the final step in denoting the inverse function. So, g-1(x) = (-8x - 4) / (3x - 7).

Therefore, the inverse function of g(x) = (7x - 4) / (3x + 8) is g-1(x) = (-8x - 4) / (3x - 7). This might seem like a lot of steps, but with practice, it becomes a straightforward process. The key is to remember to swap x and y and then solve for y. The inverse function we've found will be crucial in determining the domain and range in the next part.

(b) Stating the Domain and Range of g-1(x) in Interval Notation

Now that we've found the inverse function, g-1(x) = (-8x - 4) / (3x - 7), let's determine its domain and range. Understanding these concepts is vital for a complete understanding of the function. The domain of a function is the set of all possible input values (x-values) for which the function is defined, and the range is the set of all possible output values (y-values) that the function can produce.

Domain of g-1(x)

The domain of a rational function (a function that is a ratio of two polynomials) is all real numbers except for the values that make the denominator equal to zero. This is because division by zero is undefined. So, to find the domain of g-1(x), we need to find the values of x that make the denominator, (3x - 7), equal to zero.

  • Set the denominator equal to zero: 3x - 7 = 0
  • Solve for x: 3x = 7, so x = 7/3

This means that g-1(x) is undefined when x = 7/3. Therefore, the domain of g-1(x) is all real numbers except 7/3. In interval notation, this is written as:

  • Domain of g-1(x): (-∞, 7/3) ∪ (7/3, ∞)

This notation indicates all numbers from negative infinity up to, but not including, 7/3, and all numbers from 7/3 (not including) to positive infinity.

Range of g-1(x)

The range of the inverse function g-1(x) is closely related to the domain of the original function g(x). To find the range of g-1(x), we first need to find the domain of g(x). Remember, g(x) = (7x - 4) / (3x + 8). We follow the same process as before: find the values of x that make the denominator equal to zero.

  • Set the denominator equal to zero: 3x + 8 = 0
  • Solve for x: 3x = -8, so x = -8/3

Thus, the domain of g(x) is all real numbers except -8/3. In interval notation:

  • Domain of g(x): (-∞, -8/3) ∪ (-8/3, ∞)

The range of the inverse function, g-1(x), is the same as the domain of the original function, g(x). Therefore, the range of g-1(x) is all real numbers except -8/3. In interval notation:

  • Range of g-1(x): (-∞, -8/3) ∪ (-8/3, ∞)

In summary, we have found the domain and range of the inverse function g-1(x) using interval notation. Understanding the relationship between a function and its inverse, particularly how their domains and ranges are related, is a fundamental concept in mathematics.

Conclusion

In this article, we have successfully found the inverse function g-1(x) for the given function g(x) = (7x - 4) / (3x + 8) and determined its domain and range. We walked through the steps of swapping x and y, solving for y, and then identifying the restrictions on the input values (domain) and output values (range). This process highlights the interconnectedness of a function and its inverse. By understanding these concepts, you can tackle more complex problems involving functions and their inverses.

For a deeper understanding of inverse functions and related topics, you can explore resources like Khan Academy's Inverse Functions section, which offers comprehensive explanations and practice exercises.