Which one of the following functions is not continuous over the interval `x \in[0,5]`?
- `f(x)=\frac{1}{(x+3)^2}`
- `f(x)=\sqrt{x+3}`
- `f(x)=x^{\frac{1}{3}}`
- `f(x)=\tan \left(\frac{x}{3}\right)`
- `f(x)=\sin ^2\left(\frac{x}{3}\right)`
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Which one of the following functions is not continuous over the interval `x \in[0,5]`?
`D`
Let `a \in(0, \infty)` and `b \in R`.
Consider the function `h:[-a, 0) \cup(0, a] \rightarrow R, h(x)=\frac{a}{x}+b`.
The range of `h` is
`D`
`h(x)=(a)/(x)+b`
`text(Graph will be in the shape of a hyperbola)`
`text(Endpoint coordinates:)`
`(-a,-1+b) and (a, 1+b)`
`:.\ “Range”=(-oo,-1+b]uu[b+1,oo)`
`=>D`
Consider the function `f: [a, b) -> R,\ f(x) = 1/x`, where `a` and `b` are positive real numbers.
The range of `f` is
`D`
`f: [a, b) -> R,\ f(x) = 1/x`
`f(a) = 1/a,\ f(b) = 1/b`
`text(S) text(ince)\ a < b,`
`=>\ f(a) > f(b)`
`:.\ text(Range)\ \ f: (1/b, 1/a]`
`=> D`
The maximal domain of the function `f` is `R text(\{1})`.
A possible rule for `f` is
`A`
`text(Maximal domain of)\ \ f -> R text(\{1})`
`->\ text(all real)\ x,\ x != 1`
`:. f(x) = (x^2 – 5)/(x – 1)`
`=> A`
The graph of the function `f: D -> R,\ f(x) = (x - 3)/(2 - x),` where `D` is the maximal domain has asymptotes
`D`
`text(Use proper fraction tool on CAS:)`
`[text(CAS: propFrac) ((x – 3)/(2 – x))]`
`f(x) = -1 – 1/(x – 2)`
`:.\ text(Asymptotes:)\ \ x = 2, y = – 1`
`=> D`