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.
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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.
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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.
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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: The person next to you ate more popcorn.
Step-by-step explanation:
3 divided by 4 is 0.75, which is equivalent to 75%. 4 divided by 5 is 0.8, which is equivalent to 80%. Now that you've calculated the percentages of how much they ate, you can assume that 80 > 75 (80 is greater than 75).
If a ratio of skiers and snumboarders is 1:2, and if there are 1250 snowboarders more than skiers, it means that is that there 2500 snowboarders and 1250 skiers.
How to check it? 2500 is 1250 more than 1250 and the ratio is still 1:2 (there are two times more snowboarders than skiers). Both conditions are fulfiled, so it's correct.
It would be 30.7% because when you change a decimal to a percent, the decimal point moves 2 places to the right. For a percent to a decimal, move the decimal point 2 places to the left
Answer:
y = x*sqrt(Cx - 1)
Step-by-step explanation:
Given:
dy / dx = (x^2 + 5y^2) / 2xy
Find:
Solve the given ODE by using appropriate substitution.
Solution:
- Rewrite the given ODE:
dy/dx = 0.5(x/y) + 2.5(y/x)
- use substitution y = x*v(x)
dy/dx = v + x*dv/dx
- Combine the two equations:
v + x*dv/dx = 0.5*(1/v) + 2.5*v
x*dv/dx = 0.5*(1/v) + 1.5*v
x*dv/dx = (v^2 + 1) / 2v
-Separate variables:
(2v.dv / (v^2 + 1) = dx / x
- Integrate both sides:
Ln (v^2 + 1) = Ln(x) + C
v^2 + 1 = Cx
v = sqrt(Cx - 1)
- Back substitution:
(y/x) = sqrt(Cx - 1)
y = x*sqrt(Cx - 1)
