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Find $f(g(x))$ and $g(f(x))$ and determine whether each pair of functions $f$ and $g$ are inverses of each other.$$f(x)=-x ext { and } g(x)=x$$...

Question

Find $f(g(x))$ and $g(f(x))$ and determine whether each pair of functions $f$ and $g$ are inverses of each other.$$f(x)=-x ext { and } g(x)=x$$

Find $f(g(x))$ and $g(f(x))$ and determine whether each pair of functions $f$ and $g$ are inverses of each other. $$f(x)=-x \text { and } g(x)=x$$



Answers

Find $f(g(x))$ and $g(f(x))$ and determine whether each pair of functions $f$ and $g$ are inverses of each other. $$f(x)=-x \text { and } g(x)=x$$

To determine it. Thes functions are in versus of each other. I'm first going to find f of g of X. To do this, I'm going to substitute the G of X function into the f of X function. This is going to look like keeping the six. But instead of writing X, we will substitute the G of X function in for the X that will then become six times the fraction X over six. We can simplify this by crossing out the sixes since six divided by six just equals one. So we're left with just X. And that is what f of G of X equals. Now to continue determining if these air inverse functions of each other, we also need to find G of f of X. In this case, we're going to substitute the f of X function into the G of X function that is going to look like the f of X function substituted where the X is in the G of X function. So, as you can see, we're gonna substitute f of X for the variable. This will look like six x over six. Once again, we can cross the six is out because six divided by six just equals one. So we're left with X and that is what G of f of X equals by definition since F of G of X.

To determine if these two functions are in versus of each other. I'm first going to find f of G of X. This means I'm going to substitute the G of X function into the f of X function. This is going to look like three. But instead of X, we're going to substitute the G of X function plus eight. So, as you can see, we're going to substitute this entire G of X function right here for the X variable that's going to look like three times X minus 8/3 plus eight. Now we can cross off the threes because three divided by three cancels out and just equals one. So we're left with X minus eight plus eight negative. Eight plus eight equals zero so we can cross that off and we're just left with X. So we know F of G of X is equal to X. Next, continue confirming if these air inverse functions of each other, I'm going to find G of f of X Or, in other words, I'm going to substitute F of X into the G of X function. This is going to look like the f of X function substituted in minus eight over three. As you can see, I'm substituting this entire f of X function in for the variable that will give me three X plus a minus eight over three. Simplifying positive eight minus eight cancels out because it equals zero. So we're left with three X over three, which I can cross the threes off because three divided by three just equals one. So we're left with X, so I can say G of f of X is equal to x by.

Inverse functions. I'm first going to find f of g of X. To do this, I'm going to take the G of X function and substitute it in to the f of X function. This will look like the original f of X function, so I have the three. But instead of the X, we're going to substitute the G of X functioning, then keep the minus seven. So once again, that will look like three times. Now we're going to substitute the entire G of X function X plus three over seven, and then we still have minus seven. Next, I'm going to distribute the three into the parentheses, giving me the re X plus nine over seven minus seven. Now, to be able to combine these two pieces into one single fraction, I'm going to turn the minus seven into a fraction with the common denominator of seven. So that would be three X plus nine over seven, minus 49/7. The reason it's 49/7 is I imagine that this is 7/1, and I know I need to get a common denominator of seven. So if I multiply the denominator by seven. I also have to multiply the numerator by seven. Now, since these air two fractions with common denominators, I can subtract, which gives me three X minus 40 since positive nine minus 49 is 40 all over seven to continue checking if these air inverse functions, I'm going to do G of f of X. Now this is basically just the opposite. We're going to take the f of X function and substitute it in to the G of X function. So this will look like the F of X functions substituted in for the variable plus three over seven. Once again, that will give us instead of X. The variable will substitute f of X in so three X minus seven and then we still have the plus three and we still have the over seven simplifying. I will end up with three x and the negative seven plus three is minus four or negative for over seven Now, since I know f of G of X does not equal X and g of f of X does not equal X, I can say thes two are not inverse function

To determine if these functions are inverse functions of each other. I'm first going to find f of G of X. This means I'm going to substitute the G of X function into the f of X function where the variable is. This will look like three over. Then we'll substitute G of X in for the variable, and we'll still have the minus four. So that would look like three over. And now we're gonna substitute G of X in so three over X plus four and then we still have the minus four positive four minus four cancels out because that equals zero. So we're left with three over three over X. Now, this does look a little awkward, so you could think of this as being three divided by the Fraction three over X. Now, if you know about dividing fractions, dividing by a fraction is the same as multiplying by its reciprocal Reciprocal is when you flip a fraction upside down so that the numerator and denominator chains positions. So instead of saying three divided by three over X, you could say three times X over three. Now you see, the three is cancel out because three divided by three equals one. So we know that this just equals X so f of G of X equals X. Now, to continue deciding if these air inverse functions, I'm going to do G of f of X, which is basically the opposite of what we just did. We're going to take Activex and substitute it in to the G of X that will look like three over. And then we're gonna substitute that f of X function in plus four. So once again, that would look like three over three, divided by X minus four. And then we still have plus four. Once again, we're gonna look at this piece here the three divided by three, divided by X minus four. And remember that three divided by a fraction like three over X minus four means the same as saying three times X minus four over three. Because dividing by a fraction is the same as multiplying by its reciprocal. When I do this, I see the threes cancel out because three divided by three just equals one. So I'm left with X minus four and then I see still have the plus four from the original equation. So I have the plus four. Now I see minus four plus four that equals zero so I can cancel those out So I can say this equals X. By definition, since f of G of X equals X and G of f of X equals, X thes functions are inverse functions.


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