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The function $f$ is one-to-one. (a) Find its inverse function $f^{-1}$ and check your answer. (b) Find the domain and the range of f and $f^{-1}$.$$f(x)= rac{2 x}{3...

Question

The function $f$ is one-to-one. (a) Find its inverse function $f^{-1}$ and check your answer. (b) Find the domain and the range of f and $f^{-1}$.$$f(x)= rac{2 x}{3 x-1}$$

The function $f$ is one-to-one. (a) Find its inverse function $f^{-1}$ and check your answer. (b) Find the domain and the range of f and $f^{-1}$. $$ f(x)=\frac{2 x}{3 x-1} $$



Answers

the function $f$ is one-to-one. (a) Find its inverse function $f^{-1}$ and check your answer. (b) Find the domain and the range of f and $f^{-1} .(c)$ Graph $f, f^{-1},$ and $y=x$ on the same coordinate axes. $$ f(x)=3 x $$

Okay, so we're asked to find the inverse of the full function. So it's start rewriting RFL backs as why And now let's interchange are X enter y. So he gets This is why here in an X here and now, let's solve for X. So we get that subject one on both sides. So we get X minus. One is you could to negative 30. Why? And now we'll divide both sides by negative three. So we have that X minus one overnight. The three is equal to y, which means that f in verse of X is equal to X minus one over negative three.

So knowing that FX equals X cubed minus one, we want to find the inverse function. Have injuries of X and the concert. Doing this by recognising that f of X is something is why so y equals X cubed minus one. And to find the inverse, we can say that X equals like huge minus one By switching the X in the life. Now we just have to solve for why or f inverse of x by first adding I went to both sides, I think Q brooding both sides to get rid of the cute to basically why equals f inverse of X que brew of X plus one. And now we need to verify that half inverse of X and f of X or actual actually increases of each other so well. Start by f of f inverse of x, and this must equal X. So we begin with of que brew of experts one which is equal to Cuba of X plus one minus one, which is equal to the cube root, and the cube cancel out and we get X plus one minus one, which is equal to just X. Likewise, you repeat the same process for F in first of f of X, which is equal to f inverse of X cubed minus one equals que brew of X cubed minus one plus one, which is equal to if you cancel out the negative one and the positive one cube roots of X cubed. And we know that that is also just X. So yes, thes two are in verses of each other now to find domains and ranges. Well, we know that there are no square roots or X is in the denominator. So we know that the domain of f of X equals the range of f of f inverse of X, which equals all real numbers because there are no domain restrictions on FX. Likewise, the range of death thanks equals the domain of f inverse of X. And we know this because there are no domain restrictions on f inverse of X either. So it's all real numbers, and now we're going to graph our original function here in green. Um, our Y intercept is at negative one, and this is a cubic function, so x cubed when X is one cubed minus 10 seven. Somewhere up here anyway, this craft grows curving upwards, but we also know that it will curve downwards and they repeat the same process for by just honestly, just flipping the graph over, like so, graphing both the original and inverse functions, which are reflected over Y equals X.

Pervades the rest to find the inverse of the falling function. So we start by resetting our FX as why and now we can interchange are X and R Y to our wise No X and her ex is not what So we get the falling And now let's solve for why? Who was stuck for subjecting one about site And now we'll take both sides to the wondered palace. So we get that. Why is equal Teoh X minus one suit a wonder power. So that is we have f inverse of X is equal to fall it

So where else? It's all for inverse function. So it's done very writing effort boxes I. And now it's time to change your Exeter. Why? So we get the falling and now it's all for Why so I start by multiplying Why on both sides. So I get why X is equal to negative three. Now dividing by axe on both sides, I get the following so we get that's F in verse of X is also able to make it a three over X.


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