Which graph could represent \(y=4^x\) ?
Functions, 2ADV F2 EQ-Bank 19
- Show that the function `y = (1-e^x)/(1 + e^x)` is an odd function? (1 mark)
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- Sketch `y = (1-e^x)/(1 + e^x)`, labelling all intercepts and asymptotes. (2 marks)
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Show Answers Only
a. `\text{See Worked Solutions}`
b.
Show Worked Solution
a. `f(x) = (1-e^x)/(1 + e^x)`
| `f(-x)` | `= (1-e^(-x))/(1 + e^(-x)) xx (e^x)/(e^x)` |
| `= (e^x-1)/(e^x + 1)` | |
| `= -(1-e^x)/(1 + e^x)` | |
| `= -f(x)` |
`:. f(x)\ text(is ODD.)`
b. `y = (1-e^x)/(1 + e^x) xx (e^(−x))/(e^(−x)) = (e^(−x)-1)/(e^(−x) + 1) = 1-2/(e^(−x) + 1)`
`text(As)\ x -> ∞, \ 2/(e^(−x) + 1) -> 2, \ y -> −1`
`text(As)\ x ->-∞, \ 2/(e^(−x) + 1) -> 0, \ y -> 1`
`text(When)\ x = 0, \ y = 0`