conic section and inverse func

2013-05-02 8:14 am

1. Find the equation of the directrices of the ellipse
x^2/ 169 + y^2/ 25 = 1

2. Let f: R→R and f(x) = 2^x + 2^(-x) for x belongs to [0, infinity).
Determine whether the function has an inverse function. If it has,
find the inversse function.

回答 (1)

用戶2053

2013-05-03 12:30 am

✔ 最佳答案

1)
x²/169 + y²/25 = 1
The equation of an ellipse has the form x²/a² + y²/b² = 1.
The equation of the directrices are x = ± a²/c , where c = √(a² - b²).
Here c = √(169 - 25) = √144 = 12
Therefore the equations of directrices are x = ± 169/12.
2)f(x) = 2ˣ + 2⁻ˣ , (x ≥ 0)y = 2ˣ + 2⁻ˣ ≥ 2√(2ˣ 2⁻ˣ) = 2
y(2ˣ) = 2²ˣ + 1
2²ˣ - y(2ˣ) + 1 = 0
2ˣ = y + √(y² - 4) or y - √(y² - 4) (rejected since y - √(y² - 4) = 2⁻ˣ for x ≥ 0)
x = log₂( y + √(y² - 4) )∴ f⁻¹(x) = log₂( x + √(x² - 4) ) , (x ≥ 2)

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