If \(f(x)=\dfrac{4-e^{5 x}}{3}\), find the inverse function \(f^{-1}(x)\). (2 marks) --- 7 WORK AREA LINES (style=lined) ---
Functions, EXT1 F1 2010 HSC 3b*
Let `f(x) = e^(-x^2)`. The diagram shows the graph `y = f(x)`.
- The graph has two points of inflection.
Find the `x` coordinates of these points. (3 marks)
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- Explain why the domain of `f(x)` must be restricted if `f(x)` is to have an inverse function. (1 mark)
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- Find a formula for `f^(-1) (x)` if the domain of `f(x)` is restricted to `x ≥ 0`. (2 marks)
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- State the domain of `f^(-1) (x)`. (1 mark)
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- Sketch the curve `y = f^(-1) (x)`. (1 mark)
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Show Answers Only
- `x = +- 1/sqrt2`
- `text(There can only be 1 value of)\ y\ text(for each value of)\ x.`
- `f^(-1)x = sqrt(ln(1/x))`
- `0 <= x <= 1`
Show Worked Solution
| i. | `y` | `= e^(-x^2)` |
| `dy/dx` | `= -2x * e^(-x^2)` | |
| `(d^2y)/(dx^2)` | `= -2x (-2x * e^(-x^2)) + e ^(-x^2) (-2)` | |
| `= 4x^2 e^(-x^2)-2e^(-x^2)` | ||
| `= 2e^(-x^2) (2x^2-1)` |
`text(P.I. when)\ \ (d^2y)/(dx^2) = 0:`
| `2e^(-x^2) (2x^2-1)` | `= 0` |
| `2x^2-1` | `= 0` |
| `x^2` | `= 1/2` |
| `x` | `= +- 1/sqrt2` |
COMMENT: It is also valid to show that `f(x)` is an even function and if a P.I. exists at `x=a`, there must be another P.I. at `x=–a`.
`text(If)\ x < 1/sqrt2\ =>\ (d^2y)/(dx^2) < 0,\ \ x > 1/sqrt2\ =>\ (d^2y)/(dx^2) > 0`
`=>\ text(Change of concavity)`
`:.\ text(P.I. at)\ \ x = 1/sqrt2`
`text(If)\ x < – 1/sqrt2\ =>\ (d^2y)/(dx^2) > 0,\ \ x > – 1/sqrt2\ =>\ (d^2y)/(dx^2) < 0`
`=>\ text(Change of concavity)`
`:.\ text(P.I. at)\ \ x = -1/sqrt2`
ii. `text(In)\ f(x), text(there are 2 values of)\ y\ text(for each value of)\ x.`
`:.\ text(The domain of)\ f(x)\ text(must be restricted for)\ f^(-1) (x)\ text(to exist).`
iii. `y = e^(-x^2)`
`text(Inverse function can be written)`
| `x` | `= e^(-y^2),\ \ \ x >= 0` |
| `lnx` | `= ln e^(-y^2)` |
| `-y^2` | `= lnx` |
| `y^2` | `= -lnx` |
| `y^2` | `=ln(1/x)` |
| `y` | `= +- sqrt(ln (1/x))` |
`text(Restricting)\ \ x>=0\ \ =>\ \ y>=0`
`:. f^(-1) (x)=sqrt(ln (1/x))`
♦ Parts (iv) and (v) were poorly answered with mean marks of 39% and 49% respectively.
| iv. | `f(0) = e^0 = 1` |
`:.\ text(Range of)\ \ f(x)\ \ text(is)\ \ 0 < y <= 1`
`:.\ text(Domain of)\ \ f^(-1) (x)\ \ text(is)\ \ 0 < x <= 1`
v.
Functions, EXT1 F1 2007 HSC 6b
Consider the function `f(x) = e^x-e^(-x)`.
- Show that `f(x)` is increasing for all values of `x`. (1 mark)
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- Show that the inverse function is given by
- `f^(-1)(x) = log_e((x + sqrt(x^2 + 4))/2)` (3 marks)
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- Hence, or otherwise, solve `e^x-e^(-x) = 5`. Give your answer correct to two decimal places. (1 mark)
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Functions, EXT1 F1 2006 HSC 5b
Let `f(x) = log_e (1 + e^x)` for all `x`.
Show that `f(x)` has an inverse. (2 marks)
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Functions, EXT1 F1 2009 HSC 3a
Let `f(x) = (3 + e^(2x))/4`.
- Find the range of `f(x)`. (1 mark)
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- Find the inverse function `f^(-1) (x)`. (2 marks)
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