Images and properties of reciprocal functions Senior one elective course,

Images and properties of reciprocal functions Senior one elective course,

In general, if x corresponds to y with respect to a corresponding relation f (x) and y = f (x), then the inverse function of y = f (x) is y = f ^ - 1 (x). The condition for the existence of inverse function is that the original function must be one-to-one corresponding (i.e., the unique x corresponds to the unique y) [the property of the inverse function] (1) the image of two functions which are inverse functions to each other

What is the inverse function? What are the other functions?

Inverse function: generally, let the value range of the function y = f (x) (x ∈ a) be c. according to the relationship between X and Y in this function, X is expressed by Y, and x = f (y) is obtained. If any value of Y in C is corresponding to it by x = f (y), X in a has a unique value corresponding to it, then x = f (y) means that y is an independent variable and X is a function of dependent variable y

What is the inverse function of y = INX?

y=lnx
x=e^y
So the inverse function is y = e ^ X

If and only if the function y = x ^ 2-2ax-3 has an inverse function on the interval [1,2]?

A is less than or equal to 1
And go up
A is greater than or equal to 2
You can use recipes
y=x^2-2ax+a^2-a^2-3
y=(x-a)^2-3-a^2
I don't need to look at the second half, just look at the position of the axis of symmetry a
Because the domain of definition is [1,2], in order to have inverse function, we only need to ensure that it is monotone in [1,2]

The function f (x) = x2 + 2aX - 3 has an inverse function on the interval [1,2] if and only if () BU HUI YA!

The necessary and sufficient condition for the existence of inverse function is that y is monotone in the interval
So the parabola axis of symmetry is not in the interval
y=x^2+2ax-3=[x-(-a)]^2-a^2-3
So the axis of symmetry is x = - A
If the axis of symmetry is in the interval
Then 1

The necessary and sufficient condition for the existence of inverse function on interval [1,2] of function f (x) = x2-2ax-3 is () A. a∈（-∞，1] B. a∈[2，+∞） C. α∈[1，2] D. a∈（-∞，1]∪[2，+∞）

Analysis: the symmetry axis of ∵ f (x) = x2-2ax-3 is x = a,
⊆⊆⊆⊆⊆⊆⊆⊆⊆⊆⊆⊆⊆⊆⊆⊆⊆⊆⊆ [a, + ∞) if and only if y = f (x) has an inverse function on [1,2],
That is, a ≥ 2 or a ≤ 1
Answer: D

Necessary and sufficient conditions for the existence of inverse functions

The necessary and sufficient condition for the existence of inverse function is that the function is monotone function in the given interval, that is, one-to-one function

Function f (x) has inverse function if and only if f (x) is monotone function, right?

No, monotonicity must have an inverse function,
But there are inverse functions which are not necessarily monotonic
So monotonicity is a sufficient and unnecessary condition

If (x 2-2) is an inverse function of [x 2-2] on the interval, X 2-2 is an inverse function A. a∈（-∞，1） B. a∈[2，+∞） C. a∈（-∞，1]∪[2，+∞） D. a∈[1，2]

Analysis: the symmetry axis of ∵ f (x) = x2-2ax-3 is x = a,
The necessary and sufficient conditions for the existence of inverse functions on [1,2] are as follows:
[1,2] ⊆ (∞, a] or [1,2] ⊆ [a, + ∞)),
That is, a ≥ 2 or a ≤ 1
Therefore, C

The necessary and sufficient condition for the existence of inverse function in the interval [1,2] of the function f (x) = x ^ 2-2ax-3 is A belongs to (- infinite, 1] A belongs to [2, + infinity) A belongs to [1,2] A belongs to (- infinite, 1] and [2, + infinite]

The necessary and sufficient condition is that f (x) is a monotone function in the interval [1,2]
So the axis of symmetry is not in the open interval (1,2)
Axis of symmetry x = a
So a = 2
Pick the last one