Answer:
Step-by-step explanation:
To find the inverse function, solve for y:
![x=f(y)\\\\x=4y^4\\\\\dfrac{x}{4}=y^4\\\\\pm\sqrt[4]{\dfrac{x}{4}}=y\\\\f^{-1}(x)=\pm\sqrt[4]{\dfrac{x}{4}}](https://tex.z-dn.net/?f=x%3Df%28y%29%5C%5C%5C%5Cx%3D4y%5E4%5C%5C%5C%5C%5Cdfrac%7Bx%7D%7B4%7D%3Dy%5E4%5C%5C%5C%5C%5Cpm%5Csqrt%5B4%5D%7B%5Cdfrac%7Bx%7D%7B4%7D%7D%3Dy%5C%5C%5C%5Cf%5E%7B-1%7D%28x%29%3D%5Cpm%5Csqrt%5B4%5D%7B%5Cdfrac%7Bx%7D%7B4%7D%7D)
f(x) is an even function, so f(-x) = f(x). Then the inverse relation is double-valued: for any given y, there can be either of two x-values that will give that result.
___
A function is single-valued. That means any given domain value maps to exactly one range value. The test of this is the "vertical line test." If a vertical line intersects the graph in more than one point, then that x-value maps to more than one y-value.
The horizontal line test is similar. It is used to determine whether a function has an inverse function. If a horizontal line intersects the graph in more than one place, the inverse relation is not a function.
__
Since the inverse relation for the given f(x) maps every x to two y-values, it is not a function. You can also tell this by the fact that f(x) is an even function, so does not pass the horizontal line test. When f(x) doesn't pass the horizontal line test, f^-1(x) cannot pass the vertical line test.
_____
The attached graph shows the inverse relation (called f₁(x)). It also shows a vertical line intersecting that graph in more than one place.
Answer:
k = 12
Step-by-step explanation:
Given:
The equation 
To find:
Value of
for which the given equation has one distinct real solution.
Solution:
The given equation is a quadratic equation.
There are always two solutions of a quadratic equation.
For the equation:
to have one distinct solution:

Here,
a = 2,
b = -k and
c = 18
Putting the values, we get:

The equation becomes:

And the one root is:

Answer:
x=2
Step-by-step explanation:
6x+11-4x=15
2x+11=15
2x=4
x=2
Given :
A function , x = 2cos t -3sin t .....equation 1.
A differential equation , x'' + x = 0 .....equation 2.
To Find :
Whether the given function is a solution to the given differential equation.
Solution :
First derivative of x :

Now , second derivative :

( Note : derivative of sin t is cos t and cos t is -sin t )
Putting value of x'' and x in equation 2 , we get :
=(-2cos t + 3sin t ) + ( 2cos t -3sin t )
= 0
So , x'' and x satisfy equation 2.
Therefore , x function is a solution of given differential equation .
Hence , this is the required solution .