Finding The Inverse Of F(x) = 8x - 1 Domain Range And Graphs

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Hey everyone! Today, we're diving deep into the fascinating world of inverse functions. Specifically, we're going to tackle the function f(x) = 8x - 1. We'll walk through the process of finding its inverse, determining the domain and range of both the original function and its inverse, and then, to top it all off, we'll graph them together to visualize their relationship. It might sound like a lot, but trust me, we'll break it down step by step so it's super easy to follow. So, grab your thinking caps, and let's get started!

1. Finding the Inverse Function

So, let's kick things off by finding the inverse function of f(x) = 8x - 1. This is a crucial first step, guys, because the inverse function essentially "undoes" what the original function does. Think of it like this: if f(x) takes an input x and spits out y, then f⁻¹(x) takes that y and spits back the original x. Pretty neat, huh?

The key idea here is to swap the roles of x and y and then solve for y. This might sound a bit abstract, but it's actually a very straightforward process. Let's break it down:

  1. Replace f(x) with y: This is just a notational change to make the next steps clearer. So, we rewrite f(x) = 8x - 1 as y = 8x - 1. This helps us visualize the relationship between the input (x) and the output (y).
  2. Swap x and y: This is the heart of finding the inverse. We're essentially asking, "If the output is x, what was the input y?" Swapping gives us x = 8y - 1. This is the algebraic representation of the inverse relationship.
  3. Solve for y: Now, we need to isolate y on one side of the equation. This will give us the inverse function in the familiar y = ... form.
    • First, add 1 to both sides: x + 1 = 8y. We're trying to get y by itself, so we undo the operations that are happening to it, one step at a time.
    • Then, divide both sides by 8: (x + 1) / 8 = y. This isolates y and gives us the inverse function.
  4. Replace y with f⁻¹(x): This is just a notational change to indicate that we've found the inverse function. So, we write f⁻¹(x) = (x + 1) / 8. This is the standard notation for the inverse of f(x).

So, there you have it! The inverse function of f(x) = 8x - 1 is f⁻¹(x) = (x + 1) / 8. See? It wasn't so bad after all!

In summary, finding the inverse function involves swapping x and y and then solving for y. This process effectively reverses the operation of the original function, giving us a new function that "undoes" what the first one did.

2. Determining the Domain and Range of f and f⁻¹

Alright, now that we've found the inverse function, let's talk about domain and range. Understanding these concepts is crucial for understanding the behavior of functions. Think of the domain as the set of all possible inputs that a function can accept without causing any mathematical mayhem (like dividing by zero or taking the square root of a negative number). The range, on the other hand, is the set of all possible outputs that the function can produce.

2.1 Domain and Range of f(x) = 8x - 1

Let's start with the original function, f(x) = 8x - 1. This is a linear function, which means it's a straight line when graphed. Linear functions are super well-behaved, guys, and they don't have any restrictions on their inputs or outputs.

  • Domain of f(x): Since we can plug in any real number for x without encountering any problems, the domain of f(x) is all real numbers. We can write this in a few ways:
    • Interval notation: (-∞, ∞)
    • Set-builder notation: {x | x ∈ ℝ} (This reads as "the set of all x such that x is an element of the real numbers.")
  • Range of f(x): Similarly, since the line extends infinitely in both directions, f(x) can output any real number. So, the range of f(x) is also all real numbers.
    • Interval notation: (-∞, ∞)
    • Set-builder notation: {y | y ∈ ℝ}

2.2 Domain and Range of f⁻¹(x) = (x + 1) / 8

Now, let's consider the inverse function, f⁻¹(x) = (x + 1) / 8. Guess what? This is also a linear function! It's just a different straight line.

  • Domain of f⁻¹(x): Again, we can plug in any real number for x without any issues. There's no division by zero, no square roots of negatives, nothing to worry about. So, the domain of f⁻¹(x) is all real numbers.
    • Interval notation: (-∞, ∞)
    • Set-builder notation: {x | x ∈ ℝ}
  • Range of f⁻¹(x): And just like the original function, the inverse function can also output any real number. So, the range of f⁻¹(x) is all real numbers.
    • Interval notation: (-∞, ∞)
    • Set-builder notation: {y | y ∈ ℝ}

2.3 The Connection Between Domain and Range of f and f⁻¹

Here's a super important thing to remember about inverse functions: The domain of f is the range of f⁻¹, and the range of f is the domain of f⁻¹. This makes perfect sense if you think about it. Remember, the inverse function "undoes" the original function. So, the outputs of the original function become the inputs of the inverse function, and vice versa.

In our case, both the original function and its inverse have a domain and range of all real numbers. This isn't always the case, but it's a nice illustration of this important relationship.

Understanding domain and range is crucial for a complete understanding of a function. It tells us what inputs are allowed and what outputs are possible. And when dealing with inverse functions, remembering the reciprocal relationship between their domains and ranges is a key concept.

3. Graphing f and f⁻¹

Okay, we've found the inverse, figured out the domains and ranges, now for the fun part: graphing! Visualizing functions can really help solidify your understanding of them. We're going to graph both f(x) = 8x - 1 and f⁻¹(x) = (x + 1) / 8 on the same set of axes. This will allow us to see their relationship in a very clear way.

3.1 Graphing f(x) = 8x - 1

Let's start with the original function, f(x) = 8x - 1. This is a linear equation in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.

  • Slope (m): The slope is 8, which means for every 1 unit we move to the right along the x-axis, we move 8 units up along the y-axis. This tells us the line is quite steep.
  • Y-intercept (b): The y-intercept is -1, which means the line crosses the y-axis at the point (0, -1).

To graph the line, we can use this information:

  1. Plot the y-intercept: Plot the point (0, -1) on the coordinate plane.
  2. Use the slope to find another point: From the y-intercept, move 1 unit to the right and 8 units up. This gives us the point (1, 7).
  3. Draw a line through the two points: Connect the points (0, -1) and (1, 7) with a straight line. Extend the line in both directions, as it goes on infinitely.

3.2 Graphing f⁻¹(x) = (x + 1) / 8

Now, let's graph the inverse function, f⁻¹(x) = (x + 1) / 8. We can rewrite this as f⁻¹(x) = (1/8)x + 1/8, which is also in slope-intercept form.

  • Slope (m): The slope is 1/8, which means for every 8 units we move to the right along the x-axis, we move 1 unit up along the y-axis. This line is much less steep than the original function.
  • Y-intercept (b): The y-intercept is 1/8, which means the line crosses the y-axis at the point (0, 1/8).

To graph this line:

  1. Plot the y-intercept: Plot the point (0, 1/8) on the coordinate plane. This will be very close to the x-axis.
  2. Use the slope to find another point: From the y-intercept, move 8 units to the right and 1 unit up. This gives us the point (8, 1 + 1/8) or (8, 9/8).
  3. Draw a line through the two points: Connect the points (0, 1/8) and (8, 9/8) with a straight line. Extend the line in both directions.

3.3 The Line of Reflection: y = x

Here's the really cool part. When you graph a function and its inverse on the same set of axes, they are always reflected across the line y = x. This line acts like a mirror, and the two functions are mirror images of each other.

Graph the line y = x on your coordinate plane. This is a straight line that passes through the origin (0, 0) and has a slope of 1. You'll notice that the graphs of f(x) and f⁻¹(x) are indeed reflections of each other across this line. This is a visual confirmation that we've found the correct inverse function.

Graphing functions and their inverses is a powerful way to understand their relationship. The reflection across the line y = x is a key visual indicator that you've found the inverse correctly. And it also highlights the fundamental idea of inverse functions: they "undo" each other.

Conclusion

So, there you have it! We've successfully found the inverse of the function f(x) = 8x - 1, determined the domain and range of both functions, and graphed them to see their relationship. We've learned that the inverse function is f⁻¹(x) = (x + 1) / 8, and that both functions have a domain and range of all real numbers. We've also seen how the graphs of a function and its inverse are reflected across the line y = x.

I hope this detailed walkthrough has helped you understand the concept of inverse functions a little better. Remember, the key is to swap x and y and solve for y. And don't forget about the important connection between the domain and range of a function and its inverse. Keep practicing, and you'll be a pro at finding inverses in no time!