Inverse Functions: Which One Has A Function Inverse?

by Mei Lin 53 views

Hey guys! Let's dive into a cool math problem today: figuring out which function has an inverse that is also a function. This might sound a bit complex, but we'll break it down step by step. We're given four functions:

  • b(x)=x2+3b(x) = x^2 + 3
  • d(x)=−9d(x) = -9
  • m(x)=−7xm(x) = -7x
  • p(x)=∣x∣p(x) = |x|

Our mission, should we choose to accept it, is to determine which of these has an inverse that behaves nicely as a function itself. So, buckle up and let's get started!

Understanding Inverse Functions

Before we jump into analyzing each function, let's quickly recap what an inverse function actually is. Think of a function like a machine: you put something in (the input, often x{x}), and it spits something else out (the output, often y{y} or f(x){f(x)}). An inverse function is like a machine that reverses this process. If you put the output of the original function into its inverse, you should get back the original input. Mathematically, if f(x){f(x)} is our function and f−1(x){f^{-1}(x)} is its inverse, then:

f−1(f(x))=x{f^{-1}(f(x)) = x}

f(f−1(x))=x{f(f^{-1}(x)) = x}

This reversal is key. But here’s the catch: not every function has an inverse that is also a function. To determine if a function has a proper inverse, we use something called the horizontal line test.

The Horizontal Line Test

The horizontal line test is a visual way to check if a function has an inverse that’s also a function. Simply put, if any horizontal line intersects the graph of the function more than once, then the inverse is not a function. Why? Because if a horizontal line intersects the graph at two points, it means that two different x{x}-values map to the same y{y}-value. When you try to reverse this with the inverse function, that single y{y}-value would have to map to two different x{x}-values, violating the definition of a function (which requires each input to have only one output).

So, with this in mind, let's tackle our four functions!

Analyzing the Functions

Let's break down each function one by one to see which ones pass the horizontal line test and thus have an inverse that is also a function.

1. b(x)=x2+3{b(x) = x^2 + 3}

The first function we have is b(x)=x2+3{b(x) = x^2 + 3}. This is a quadratic function, which means its graph is a parabola. Specifically, it’s a parabola that opens upwards, with its vertex (the lowest point) at (0,3){(0, 3)}. Now, picture this parabola in your mind, or even better, sketch it out on a piece of paper. If you draw a horizontal line through this parabola—say, at y=4{y = 4}—you’ll notice that it intersects the parabola at two points. This is because parabolas are symmetrical around their vertex. For instance, both x=1{x = 1} and x=−1{x = -1} would give you b(x)=4{b(x) = 4}.

Because the horizontal line test fails (the line intersects the graph more than once), the inverse of b(x)=x2+3{b(x) = x^2 + 3} is not a function. To see why this happens mathematically, you can try to find the inverse. Replace b(x){b(x)} with y{y}, so you have y=x2+3{y = x^2 + 3}. To find the inverse, swap x{x} and y{y} and solve for y{y}:

x=y2+3{x = y^2 + 3}

y2=x−3{y^2 = x - 3}

y=±x−3{y = \pm\sqrt{x - 3}}

Notice the ±{\pm}? This means for a single value of x{x}, you get two values of y{y}, which confirms that the inverse is not a function. It's like a machine that takes one input and gives you two different outputs – a big no-no in Functionland! Therefore, b(x){b(x)} does not have an inverse that is a function.

2. d(x)=−9{d(x) = -9}

Next up, we have d(x)=−9{d(x) = -9}. This is a constant function. No matter what x{x} you put in, the output is always −9{-9}. The graph of this function is a horizontal line at y=−9{y = -9}. Now, let's apply the horizontal line test. If we draw any horizontal line other than y=−9{y = -9}, it won't intersect the graph at all. However, if we draw the line y=−9{y = -9}, it coincides perfectly with the graph, meaning it intersects infinitely many times! This is a dramatic fail for the horizontal line test.

To understand this mathematically, let's try to find the inverse. If y=−9{y = -9}, swapping x{x} and y{y} gives us x=−9{x = -9}. Now, try to solve for y{y}. You can't! There's no y{y} left in the equation. The inverse would have to take the input −9{-9} and give back every possible x{x} value, which is simply not a function. It's a function's worst nightmare – chaos and infinitely many outputs for one input! Thus, d(x)=−9{d(x) = -9} does not have an inverse that is a function.

3. m(x)=−7x{m(x) = -7x}

Our third contender is m(x)=−7x{m(x) = -7x}. This is a linear function with a slope of −7{-7} and a y{y}-intercept of 0{0}. The graph is a straight line that slopes downwards from left to right. Now, let's visualize the horizontal line test. Can you imagine drawing any horizontal line that intersects this graph more than once? Nope! A straight line will only be intersected once by any horizontal line (except for a horizontal line coinciding with the function itself, which isn't the case here). This is a good sign!

To confirm this mathematically, let's find the inverse. Start with y=−7x{y = -7x}. Swap x{x} and y{y}:

x=−7y{x = -7y}

Now, solve for y{y}:

y=−x7{y = -\frac{x}{7}}

We've found the inverse function: m−1(x)=−x7{m^{-1}(x) = -\frac{x}{7}}. This is also a linear function, which means it's a well-behaved function in its own right. It takes one input and produces exactly one output. Success! Therefore, m(x)=−7x{m(x) = -7x} does have an inverse that is a function.

4. p(x)=∣x∣{p(x) = |x|}

Last but not least, we have p(x)=∣x∣{p(x) = |x|}, the absolute value function. This function takes any input and returns its non-negative value. The graph of p(x)=∣x∣{p(x) = |x|} is a V-shape, with the point of the V at the origin (0,0){(0, 0)}. The left side of the V slopes downwards, and the right side slopes upwards.

Now, let's apply the horizontal line test. Imagine drawing a horizontal line at, say, y=2{y = 2}. It intersects the V at two points: x=2{x = 2} and x=−2{x = -2}. This means both p(2)=2{p(2) = 2} and p(−2)=2{p(-2) = 2}. The horizontal line test fails once again!

To see this mathematically, let's try finding the inverse. Start with y=∣x∣{y = |x|}. Swap x{x} and y{y}:

x=∣y∣{x = |y|}

Now, try to solve for y{y}. If x{x} is positive, then y{y} could be either x{x} or −x{-x}. For example, if x=3{x = 3}, then y{y} could be 3{3} or −3{-3}. This means the inverse is not a function because one input value gives you two possible output values. It’s a double-trouble situation! Thus, p(x)=∣x∣{p(x) = |x|} does not have an inverse that is a function.

Conclusion

Alright, guys, we've made it through all four functions! After carefully analyzing each one using the horizontal line test and a bit of algebraic maneuvering, we've found our winner. The only function from the list that has an inverse that is also a function is:

m(x)=−7x{m(x) = -7x}

The linear function m(x)=−7x{m(x) = -7x} passes the horizontal line test and has a well-defined inverse function, m−1(x)=−x7{m^{-1}(x) = -\frac{x}{7}}. The other functions—b(x)=x2+3{b(x) = x^2 + 3}, d(x)=−9{d(x) = -9}, and p(x)=∣x∣{p(x) = |x|}—do not have inverses that are functions due to failing the horizontal line test.

So, there you have it! We've successfully navigated the world of inverse functions and identified the one that behaves just right. Keep practicing, and you'll become a master of inverse functions in no time!