Verify that the function f^-1(x) is the inverse of f(x) by...

Verify that the function $f^{-1}(x)$ is the inverse of $f(x)$ by showing that $f\left(f^{-1}(x)\right)=x$ and $f^{-1}(f(x))=x .$ Graph $f(x)$ and $f^{-1}(x)$ on the same axes to show the symmetry about the line $y=x$ $$f(x)=\frac{x+3}{x+4}, x \neq-4 ; f^{-1}(x)=\frac{3-4 x}{x-1}, x \neq 1$$

Verify that the function $f^{-1}(x)$ is the inverse of $f(x)$ by showing that $f\left(f^{-1}(x)\right)=x$ and $f^{-1}(f(x))=x .$ Graph $f(x)$ and $f^{-1}(x)$ on the same axes to show the symmetry about the line $y=x$
$$f(x)=\frac{x+3}{x+4}, x \neq-4 ; f^{-1}(x)=\frac{3-4 x}{x-1}, x \neq 1$$

Key Concepts

Inverse Functions

An inverse function reverses the input-output relationship of the original function. If f is a function that maps an element x to y, then the inverse function f?¹ maps y back to x. This concept is crucial because it allows us to 'undo' the effect of the original function, thereby providing a method to solve equations where the function is involved.

Function Composition

Function composition involves applying one function to the result of another function. In the context of inverse functions, composing a function with its inverse (in either order) should yield the identity function, meaning that f(f?¹(x)) = x and f?¹(f(x)) = x. This property is an essential method for verifying that two functions are indeed inverses of each other.

Graphical Symmetry About the Line y = x

The graphs of a function and its inverse are mirror images of each other across the line y = x. This symmetry provides a visual confirmation of the inverse relationship between the functions. By plotting both graphs on the same set of axes, one can observe whether they reflect correctly about the line y = x, which serves as a geometric check of the inverse property.

Domain and Range Considerations

The definitions of a function and its inverse depend critically on their domains and ranges. When verifying inverse functions, it is important to ensure that the domain of the original function corresponds to the range of its inverse, and vice versa. Restrictions on the domain, such as values that would cause division by zero, must be carefully identified and respected.

Transcript

00:01 Hey guys, so before verifying that these two are inferences of each other, i want you guys to pay attention to the two rational functions here.

00:09 So again, because there are ratios, right? so there will be potentially horizontal and vertical asymptotes.

00:16 So the values that the function cannot be.

00:18 Okay, so the vertical asymptotes are already given here, but what about the horizontal asymptote? so look at the original function f of x, or since the top and the bottom have the same degree of one, right? then the horizontal asymptote should be why different from the ratio of the leading coefficient, 1 over 1, which is why different from 1? okay, now let's look at the inverse function here.

00:45 So same thing.

00:46 Why cannot be the ratio of the leading coefficients, which is negative 4 over 1, which is negative 4.

00:53 So i want you guys to pay attention to all the horizontal and vertical asymptotes so that it will be easier for you guys to graph later on.

01:01 Okay, awesome.

01:04 So now to verify, let's do f of f inverse and f inverse of fx.

01:10 Okay, so for the first function here, remember that we are inputting the inverse function here, not x.

01:26 Be careful, just like this.

01:31 All right, so let's simplify here.

01:39 So for the top, right, multiply x minus 1 to both the top and the bottom here.

01:48 Right, so that we can have the same denominator.

01:53 And then at the bottom here, let's do the same thing.

02:04 All right.

02:10 So now that we have the same denominator, right, we can just combine our numerators together, just like this.

02:18 And i will rewrite this as a division.

02:22 So you guys can visualize it better.

02:29 Okay, so keep in mind that this is just review of everything that we've learned before...

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