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Merge branch 'main' into v0.16

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jverzani 2023-04-10 07:14:11 -04:00
commit dcebed6fdd
2 changed files with 12 additions and 12 deletions
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16
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 $g_2(x)=f_2^{-1}(x)$?

8
quarto/precalc/transformations.qmd
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@ -94,7 +94,7 @@ plot!(gf, label="g∘f")
:::{.callout-note}
## Note
Unlike how the basic arithmetic operations are treated, `Julia` defines the infix Unicode operator `\\circ[tab]` to represent composition of functions, mirroring mathematical notation. This infix operations takes in two functions and returns an anonymous function. It can be useful and will mirror standard mathematical usage up to issues with precedence rules.
Unlike how the basic arithmetic operations are treated, `Julia` defines the infix Unicode operator `\circ[tab]` to represent composition of functions, mirroring mathematical notation. This infix operations takes in two functions and returns an anonymous function. It can be useful and will mirror standard mathematical usage up to issues with precedence rules.
:::
@ -108,7 +108,7 @@ $$
(f \circ g)(x) = (e^x - x)^2 + 2(e^x - x) - 1.
$$
If can be helpful to think of the argument to $f$ as a "box" that gets filled in by $g$:
It can be helpful to think of the argument to $f$ as a "box" that gets filled in by $g$:
@ -484,7 +484,7 @@ If $h(x) = (f \circ g)(x) = \sin^2(x)$ which is a possibility for $f$ and $g$:
#| hold: true
#| echo: false
choices = [raw"``f(x)=x^2; \quad g(x) = \sin^2(x)``",
raw"```f(x)=x^2; \quad g(x) = \sin(x)``",
raw"``f(x)=x^2; \quad g(x) = \sin(x)``",
raw"``f(x)=\sin(x); \quad g(x) = x^2``"]
answ = 2
radioq(choices, answ)
@ -519,7 +519,7 @@ Let $h(x) = 4x^2$ and $f(x) = x^2$. Which is **not** true:
#| echo: false
choices = [L"The graph of $h(x)$ is the graph of $f(x)$ stretched by a factor of ``4``",
L"The graph of $h(x)$ is the graph of $f(x)$ scaled by a factor of ``2``",
L"The graph of $h(x)$ is the graph of $f(x) shifted up by ``4`` units"]
L"The graph of $h(x)$ is the graph of $f(x)$ shifted up by ``4`` units"]
answ = 3
radioq(choices, answ)
```