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Update inversefunctions.qmd

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some typos.
This commit is contained in:
Fang Liu 2023-04-10 11:40:13 +08:00
parent 9ac2fe9286
commit 4dc62406cf
1 changed files with 8 additions and 8 deletions
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quarto/precalc/inversefunctions.qmd
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@ -175,7 +175,7 @@ Consider the function $f(x) = (1+x^2)^{-1}$. This bell-shaped function is even (
y &= \frac{1}{1 + x^2}\\
1+x^2 &= \frac{1}{y}\\
x^2 &= \frac{1}{y} - 1\\
x &= \sqrt{(1-y)/y}, \quad 0 \leq y \leq 1.
x &= \sqrt{(1-y)/y}, \quad 0 < y \leq 1.
\end{align*}
@ -322,7 +322,7 @@ Is it possible that a function have two different inverses?
```{julia}
#| hold: true
#| echo: false
choices = [L"No, for all $x$ in the domain an an inverse, the value of any inverse will be the same, hence all inverse functions would be identical.",
choices = [L"No, for all $x$ in the domain and an inverse, the value of any inverse will be the same, hence all inverse functions would be identical.",
L"Yes, the function $f(x) = x^2, x \geq 0$ will have a different inverse than the same function $f(x) = x^2, x \leq 0$"]
answ = 1
radioq(choices, answ)
@ -337,7 +337,7 @@ A function takes a value $x$ adds $1$, divides by $2$, and then subtracts $1$. I
```{julia}
#| hold: true
#| echo: false
choices = [L"Yes, the function is the linear function $f(x)=(x+1)/2 + 1$ and so is monotonic.",
choices = [L"Yes, the function is the linear function $f(x)=(x+1)/2 - 1$ and so is monotonic.",
L"No, the function is $1$ then $2$ then $1$, but not \"one-to-one\""
]
answ = 1
@ -571,8 +571,8 @@ Find the inverse function of $f(x) = (x^3 + 4)/5$.
#| hold: true
#| echo: false
choices = [
"``f^{-1}(x) = (5y-4)^{1/3}``",
"``f^{-1}(x) = (5y-4)^3``",
"``f^{-1}(x) = (5x-4)^{1/3}``",
"``f^{-1}(x) = (5x-4)^3``",
"``f^{-1}(x) = 5/(x^3 + 4)``"
]
answ = 1
@ -686,7 +686,7 @@ The function $f(x) = (ax + b)/(cx + d)$ is known as a [Mobius](http://tinyurl.co
* and $f_4(x) = x + a/c$ is a translation.
For $x=10$, what is $f(10)$?
For $x=10$, $a=1$, $b=2$, $c=3$ and $d=5$, what is $f(10)$?
```{julia}
@ -713,7 +713,7 @@ The last two answers should be the same, why?
#| hold: true
#| echo: false
choices = [
L"As $f_4(f_3(f_2(f)_1(x))))=(f_4 \circ f_3 \circ f_2 \circ f_1)(x)$",
L"As $f_4(f_3(f_2(f_1(x))))=(f_4 \circ f_3 \circ f_2 \circ f_1)(x)$",
L"As $f_4(f_3(f_2(f_1(x))))=(f_1 \circ f_2 \circ f_3 \circ f_4)(x)$",
"As the latter is more complicated than the former."
]
@ -721,7 +721,7 @@ answ=1
radioq(choices, answ)
```
Let $g_1$, $g_2$, $g_3$, and $g_4$ denote the inverse functions. Clearly, $g_1(x) = x- d/c$ and $g+4(x) = x - a/c$, as the inverse of adding a constant is subtracting the constant.
Let $g_1$, $g_2$, $g_3$, and $g_4$ denote the inverse functions. Clearly, $g_1(x) = x- d/c$ and $g_4(x) = x - a/c$, as the inverse of adding a constant is subtracting the constant.
What is $g_2(x)=f_2^{-1}(x)$?