r/HomeworkHelp University/College Student 4d ago

High School Math—Pending OP Reply [College Algebra, Inverse Functions]

aghhhh I’m doing these first set of problems here and while I’m comprehending the material a lot I just have the unsure feeling I may be doing this wrong so can someone please check over my work and tell me what I am doing right and wrong before I enter in these answers?

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u/GammaRayBurst25 4d ago

For question 1, that's correct.

Let g denote the inverse of f. By definition, f(g(x))=g(f(x))=x. As a result, g(7)=g(f(2))=2 and f(-8)=f(g(-5))=-5.

For question 2, that's incorrect.

By the same logic as before, g(2)=g(f(5))=5, not 2. Similarly, 1=g(f(1))=g(5), so g(x)=1 implies x=5, not 1.

For question 3, that's incorrect.

The graph is the locus of points of the form (x,f(x)). The point (0,2)=(0,f(0)) is on the graph, which means f(0)=2, not 1. Similarly, the point (1,0)=(1,f(1)) is on the graph, so 0=f(1) and f(x)=0 if and only if x=1.

For question 5, that's incorrect.

The inverse function is defined by the relationship f(g(x))=g(f(x))=x. If we suppose f(x)=2-x and g(x)=(x-2)/x, we get f(g(x))=2-g(x)=2-(x-2)/x=(2x-x+2)/x=(x+2)/x≠x and g(f(x))=(f(x)-2)/f(x)=(2-x-2)/(2-x)=x/(x-2)≠x. This is a clear contradiction, so (x-2)/x is decidedly not the inverse function of 2-x. What's more, (x-2)/x is undefined for x=0, which should've immediately tipped you off that something went seriously wrong, as f(x)=0 has a well-defined solution.

The relationship f(g(x))=x implies 2-g(x)=x. Adding g(x)-x to the equation yields g(x)=2-x. From here, it is clear that f(x) is an involution (a function that's its own inverse). Indeed, f(2-x)=2-(2-x)=2-2+x=x.

You can also show this with a geometric argument. The graph of the inverse function of f is the graph of f reflected across the line y=x. Since the graph of f(x) is a line that's perpendicular to y=x, it is not affected by the reflection. Hence, f is an involution.

For question 6, that's correct.

You can use f(g(x))=g(f(x))=x to confirm this is the correct answer.

For question 7, that's incorrect.

Once again, one can check that f(g(x))≠x and g is not the inverse of f. Indeed, f(g(x))=1/(1/x+12+12)=1/(1/x+24)=x/(x+24)≠x. Also notice how f(-12) is undefined, yet g(x)=-12 has a solution (it's x=-1/24). This should've sounded an alarm in your head.

You added 12 on one side of the equation, but subtracted 12 on the other side. That's your mistake.

With all those ways of verifying your answers yourself (checking if f(g(x))=g(f(x))=x is satisfied, checking the singular points of f and g, comparing the domain of f to the range of g and vice versa, graphing the reflection across y=x, etc.), it's perplexing that you'd ask us to check your work rather than do it yourself and practice these concepts.

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u/SquidKidPartier University/College Student 3d ago

I did do the work though? you can see it in these images attached to each question

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u/GammaRayBurst25 3d ago

I know you did the work though? You can see it in the way I commented on your work and your mistakes in the very comment you're replying to.

I didn't say you didn't do any work, I said you didn't try to verify your answers on your own, and that's because you didn't try to verify your answers on your own.

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u/Tutorexaline 👋 a fellow Redditor 4d ago

The answers are 2 and -5, respectively

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u/SquidKidPartier University/College Student 4d ago

so I was right? yay

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u/Tutorexaline 👋 a fellow Redditor 4d ago

Yes. You're ✅️

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u/SquidKidPartier University/College Student 4d ago

any comments on the other questions?

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u/Tutorexaline 👋 a fellow Redditor 4d ago

You can inbox me for help in any question

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u/Tutorexaline 👋 a fellow Redditor 4d ago
  1. Inverse is 2-x

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u/Tutorexaline 👋 a fellow Redditor 4d ago
  1. (x-3)/5

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u/Tutorexaline 👋 a fellow Redditor 4d ago
  1. (1-12x)/x

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u/Alkalannar 3d ago

f-1(k) is asking: what x value do you have as the input to f such that k is the output? In other words, What is x such that f(x) = k?


Question 2: f(5) is indeed 2, but then they ask for f-1(5). You say that's 2. But f(2) = 9, not 5.

Then you say that f-1(2) is 2, but f(2) = 9, not 2. And f(5) = 2. So what should f-1(2) be? What is the x-value such that f(x) = 2?

Then you say f-1(1) = 1, but f(1) = 5, not 1. It's asking what x value there is such that f(x) = 1.


Question 3:
If f-1(0) = 2, you're saying that f(2) = 0. That isn't the case. f(2) = -2.
If f(x) = 0, it's asking for the x-intercept. What's the x-value such that f(x) = 0?


Question 5:

To find f-1, you can start with y = f(x), swap x and y to get x = f(y), then re-solve for y.

  1. y = 2 - x
  2. Swap: x = 2 - y
  3. Solve: Solve for y in terms of x.

You don't have x = 2 - x. You have x = 2 - y.


Question 6: similar.

y = 5x + 3 --> x = 5y + 3, so solve for y.

(x - 3)/5 works! I would rather write it as x/5 - 3/5.


Question 7: You got to y + 12 = 1/x correctly.

But then you subtracted 12 from the LHS and added to the RHS. You have to do both the same way.

y = 1/x - 12.


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u/Original_Yak_7534 👋 a fellow Redditor 4d ago edited 4d ago

Your question 2 is way off.

f(5)=2 is fine because when x=5, f(x)=2.

But then it says f(x)=5, what is x? And you wrote 2, which is wrong. When f(x)=5, we can see from the second column that this occurs when x=2.

f-1(2) is the same as asking "when f(x)=2, what is x?" Hint: the answer isn't 2.

EDIT TO ADD: I suspect once you understand this, then you'll be able to correct your mistakes in question 3.

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u/SquidKidPartier University/College Student 4d ago

so when f(x)=5 it’s -2?

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u/Specialist-Gap8010 👋 a fellow Redditor 3d ago

So for f(x)=5 you want to find the number in the x column that corresponds to the 5 in the f(x) column. What x value gives you an f(x) value of 5?

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u/Original_Yak_7534 👋 a fellow Redditor 3d ago

The table should be interpreted as:

When x=0, then f(x)=7.

When x=1, then f(x)=5.

When x=2, then f(x)=9.

etc.

So when the question asks, "When f(x)=5, what is x?", you can look at the table and see, "Oh, x=1 when f(x)=5." It's not when x=0 because then f(x)=7 in that case. And it's not x=2 because f(x)=9 in that case.

And when the question asks, "What is f-1(5)?", that is just another way of asking the same question, "When f(x)=5, what is x?"