True or False:
For a trigonometric function, y = f(x), then x = F^-1(y). Explain your answer. (the capital F denotes function, not relation)
For a one-to-one function, y = f(x), then x = f^-1(y). Explain your answer.
For any function, x = f^-1(y), then y = f(x). Explain your answer.
Stacey Warren - Expert brainly.com
Hey! We 've verified this expert answer for you, click below to unlock the details :)
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga.
Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus.
Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
for a function
we can say
\[\iff\](this symbol means if and only if, it's nice to remember some mathematical symbols on the go!)
is one-to-one AND onto
both conditions must be satisfied simultaneously
Now what this means?
A function is said to be one-to-one(or one-one), if a number in the range of the function is only linked to a specific number in the domain
Think of it this way, if you were to find inverse of a number from the range of f(x), if it were linked to multiple numbers, then it would not be an inverse function, because a function by definition takes an input and gives only 1 output
consider the drawing:
Now example of an one-one function:
Onto is when EACH number in the range has a link to a number in domain
Both drawing 1 and 2 above are not onto because there are elements which have no links back to set x, (eg. v)
Think of it like this in drawing 2
if you were to calculate the inverse at v, it would not be defined as v is not linked to some number in set x, thus your inverse function is not defined, so a function must be onto
Think about these points and try to attempt the question
So y=f(x) can only be inverted to x=f^-1 (y) when f(x) is one to one? Here's what I have so far, but I'm very unsure about it.
1. y = f(x), then x = F^-1(y) is true because capital F requires the equation to be a one to one function, not a relation.
2. y = f(x), then x = f^-1(y). True, because it's stated in the question that it is a one to one function and thus can be inverted.
3. x = f^-1(y), then y = f(x). False, because we do not know whether it is a function or relation and cannot determine if it can be inverted.
y=f(x) is invertable if and only if f(x) is one-one and onto
Not the answer you are looking for? Search for more explanations.
example of a function whose inverse can exist:
for example consider the sine function
At x=pi/4 and 3pi/4 we have
Now this shows that the sine function is not one-one,
if we want to evaluate
we are getting 2 answers pi/4 and 3pi/4, but a function only gives 1 answer
therefore for inverse trigonometric functions, we have to apply certain restrictions about which you would study later if you've already not studied it
thus for your question 1)
trig functions are not one-one so that already tells you their inverse cannot exist unless we apply some restrictions
so it is FALSE
For your question 2)
A function must be both one-one and onto, since the question says function is one one that's 1 condition satisfied, since it's not given that the function is onto, we assume it to be non-onto therefore inverse does not exist
for your question 3)
False, not every function has an inverse, some functions are not one-one, some are not onto and some are neither.
Ah, my textbook only covered the definition of one-to-one, it didn't even mention onto! Thank you so much for explaining it! So we basically assume unless stated that a function fulfills both conditions, it cannot be inverse? I hope I got it now!
Yes, a function must satisfy both conditions
of one-one and onto
I hope you understood the sine example, the reason why trig functions are not one-one
Since your textbook only covers one-one, you may consider Q2 to be as true actually(as per your textbook)
A function that is one-one is also referred to as injective function
and a function that is onto is also referred to as surjective function
So inverse exists for a function that is both injective and surjective
you can read more about it here: