work on better figures
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jverzani
@@ -42,6 +42,101 @@ For these examples, the domain of both $f(x)$ and $g(x)$ is all real values of $
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In general the range is harder to identify than the domain, and this is the case for these functions too. For $f(x)$ we may know the $\cos$ function is trapped in $[-1,1]$ and it is intuitively clear than all values in that set are possible. The function $h(x)$ would have range $[0,\infty)$. The $s(x)$ function is either $-1$ or $1$, so only has two possible values in its range. What about $g(x)$? It is a parabola that opens upward, so any $y$ values below the $y$ value of its vertex will not appear in the range. In this case, the symmetry indicates that the vertex will be at $(1/2, -1/4)$, so the range is $[-1/4, \infty)$.
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::: {#fig-domain-range layout-nrow=2}
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```{julia}
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#| echo: false
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plt = let
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gr()
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# domain/range shade
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λ = 1.2
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a, b = .1, 3
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f(x) = (x-1/2) + sin((x-1)^2)
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empty_style = (xaxis=([], false),
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yaxis=([], false),
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framestyle=:origin,
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legend=false)
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axis_style = (arrow=true, side=:head, line=(:gray, 1))
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text_style = (10,)
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fn_style = (;line=(:black, 3))
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fn2_style = (;line=(:red, 4))
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mark_style = (;line=(:gray, 1, :dot))
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domain_style = (;fill=(:orange, 0.35))
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range_style = (; fill=(:blue, 0.35))
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xs = range(a,b, 1000)
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y0,y1 = extrema(f.(xs))
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Δy = (y1-y0)/60
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Δx = (b - a)/75
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plot(; aspect_ratio=:equal, empty_style...)
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plot!([-.25,3.25],[0,0]; axis_style...)
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plot!([0,0], [min(-2Δy, y0 - Δy), y1 + 4Δy]; axis_style... )
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plot!(f, a, b; fn_style...)
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plot!(Shape([a,b,b,a], Δy * [-1,-1,1,1] ); domain_style...)
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plot!(Shape(Δx*[-1,1,1,-1], [y0,y0,y1,y1]); range_style...)
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plot!([a,a], [0, f(a)]; mark_style...)
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plot!([b,b], [0, f(b)]; mark_style...)
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plot!([a, b], [f(a), f(a)]; mark_style...)
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plot!([a, b], [f(b), f(b)]; mark_style...)
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end
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plot
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```
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```{julia}
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#| echo: false
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plt = let
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a, b = 0, 2pi
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λ = 1.1
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Δx, Δy = .033, .1
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δx = 0.05
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f(x) = sec(x)
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g(x) = abs(f(x)) < 5 ? f(x) : NaN
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empty_style = (xaxis=([], false),
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yaxis=([], false),
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framestyle=:origin,
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legend=false)
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axis_style = (arrow=true, side=:head, line=(:gray, 1))
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text_style = (10,)
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fn_style = (;line=(:black, 3))
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fn2_style = (;line=(:red, 4))
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mark_style = (;line=(:gray, 1, :dot))
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domain_style = (;fill=(:orange, 0.35))
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range_style = (; fill=(:blue, 0.35))
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plot(; empty_style...)
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plot!(g, a+0.1, b; fn_style...)
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plot!(λ*[a,b],[0,0]; axis_style...)
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plot!([0,0], λ*[-5,5+1/2]; axis_style...)
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vline!([pi/2, 3pi/2]; line=(:gray, :dash))
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plot!(Shape([a,a+pi/2-δx,a+pi/2-δx,a], Δy*[-1,-1,1,1]); domain_style...)
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plot!(Shape([a+pi/2+δx, a+pi/2+pi-δx,a+pi/2+pi-δx,a+pi/2+δx],
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Δy*[-1,-1,1,1]); domain_style...)
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plot!(Shape([3pi/2 + δx, 2pi, 2pi, 3pi/2+δx],
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Δy*[-1,-1,1,1]); domain_style...)
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plot!(Shape(Δx*[-1,1,1,-1], [-5, -5,-1,-1]); range_style...)
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plot!(Shape(Δx*[-1,1,1,-1], [ 5, 5,1,1]); range_style...)
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end
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plot
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```
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```{julia}
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#| echo: false
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plotly()
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nothing
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```
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The top figure shows the domain and range of the function as highlighted intervals. The bottom figure shows that the domain may be a collection of intervals. (In this case the $\sec$ function is not defined at $\pi/2 + k \pi$ for integer $k$) and the range may be a collection of intervals. (In this case, the $\sec$ function never have a value in $(-1,1)$.
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:::
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:::{.callout-note}
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## Note
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@@ -90,7 +185,7 @@ Whereas function definitions and usage in `Julia` mirrors standard math notation
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:::
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### The domain of a function
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### The domain of a function in Julia
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Functions in `Julia` have an implicit domain, just as they do mathematically. In the case of $f(x)$ and $g(x)$, the right-hand side is defined for all real values of $x$, so the domain is all $x$. For $h(x)$ this isn't the case, of course. Trying to call $h(x)$ when $x < 0$ will give an error:
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@@ -347,10 +442,27 @@ In our example, we see that in trying to find an answer to $f(x) = 0$ ( $\sqrt{2
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```{julia}
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#| echo: false
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plot(q, a, b, linewidth=5, legend=false)
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plot!(zero, a, b)
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plot!([a, b], q.([a, b]))
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scatter!([c], [q(c)])
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plt = let
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gr()
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plt = plot(q, a, b, linewidth=5, legend=false)
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plot!(plt, zero, a, b)
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plot!(plt, [a, b], q.([a, b]))
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scatter!(plt, [c], [q(c)]; marker=(:circle,))
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scatter!(plt, [a,b], [q(a), q(b)]; marker=(:square,))
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annotate!(plt, [
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(a, 0, text(L"a", 10,:top)),
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(b, 0, text(L"b", 10, :top)),
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(c, 0, text(L"c", 10, :bottom))
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])
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end
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plt
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```
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```{julia}
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#| echo: false
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plotly()
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nothing
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```
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Still, `q(c)` is not really close to $0$:
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@@ -398,6 +510,7 @@ c = secant_intersection(f, a, b)
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p = plot(f, a, b, linewidth=5, legend=false)
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plot!(p, zero, a, b)
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scatter!([a,b], [f(a), f(b)]; marker=(:square,))
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plot!(p, [a,b], f.([a,b]));
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scatter!(p, [c], [f(c)])
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@@ -918,6 +1031,86 @@ answ = 1
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radioq(choices, answ)
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```
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##### Question
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::: {#fig-floor-function}
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```{julia}
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#| echo: false
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plt = let
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gr()
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empty_style = (xaxis=([], false),
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yaxis=([], false),
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framestyle=:origin,
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legend=false)
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axis_style = (arrow=true, side=:head, line=(:gray, 1))
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text_style = (10,)
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fn_style = (;line=(:black, 3))
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fn2_style = (;line=(:red, 4))
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mark_style = (;line=(:gray, 1, :dot))
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domain_style = (;fill=(:orange, 0.35), line=nothing)
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range_style = (; fill=(:blue, 0.35), line=nothing)
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ts = range(0, 2pi, 100)
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xys = sincos.(ts)
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xys = [.1 .* xy for xy in xys]
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plot(; empty_style..., aspect_ratio=:equal)
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plot!([-4.25,4.25], [0,0]; axis_style...)
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plot!([0,0], [-4.25, 4.25]; axis_style...)
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for k in -4:4
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P,Q = (k,k),(k+1,k)
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plot!([P,Q], line=(:black,1))
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S = Shape([k .+ xy for xy in xys])
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plot!(S; fill=(:black,))
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S = Shape([(k+1,k) .+ xy for xy in xys])
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plot!(S; fill=(:white,), line=(:black,1))
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end
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current()
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end
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plotly()
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plt
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```
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The `floor` function rounds down. For example, any value in $[k,k+1)$ rounds to $k$ for integer $k$.
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:::
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The figure shows the `floor` function which is useful in programming. It rounds down to the first integer value.
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From the graph, what is the domain of the function?
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```{julia}
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#| hold: true
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#| echo: false
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choices = [
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"The entire real line",
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"The entire real line except for integer values",
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"The integers"
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]
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answer = 1
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radioq(choices, answ)
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```
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From the graph, what is the range of the function?
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```{julia}
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#| hold: true
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#| echo: false
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choices = [
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"The entire real line",
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"The entire real line except for integer values",
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"The integers"
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]
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answer = 3
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radioq(choices, answ)
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```
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(This graphic uses the convention that a filled in point is present, but an open point is not, hence each bar represents some $[k, k+1)$.)
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###### Question
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