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The definition domain of function f [x] is R, satisfying that f[x+1]=2f[x],

and when x belongs to (0, 1], f [x]=x(x-1).

If for any x belongs to (-[Infinity], m], there is f [x]>=-8/9, find the value range of m

  1. How to draw an image of the function f [x] within a certain interval

2.How to calculate the value of m?

The image roughly looks like this:

enter image description here

Clear["Global`*"]
f[x_] := 2^Floor[x]  Mod[x, 1]  (Mod[x, 1] - 1)
Plot[f[x], {x, 0, 4}]
Grid[Table[{n < x < n + 1, 
   Expand@FullSimplify[f[x], Assumptions -> n < x < n + 1]}, {n, -5, 
   5}], Frame -> All]
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3 Answers 3

8
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Edit

  • For x<0,we rewrite f[x+1]=2f[x] to `f[x]=f[x+1]/2;
Clear["Global`*"];
f[x_] := x (x - 1); /; 0 < x <= 1;
f[x_] := 2 f[x - 1] /; x >= 1;
f[x_] := f[x + 1]/2 /; x < 0
Plot[f[x], {x, -3, 3}]

enter image description here

Original

f[x_] := x (x - 1); /; 0 < x <= 1;
f[x_] := 2 f[x - 1] /; x >=1;
Plot[f[x], {x, 0, 3}]

enter image description here

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3
  • $\begingroup$ If for any x belongs to (-[Infinity], m], there is f [x]>=-8/9, find the value range of m $\endgroup$
    – csn899
    Nov 11, 2023 at 2:12
  • $\begingroup$ How can I draw an image when x<0? $\endgroup$
    – csn899
    Nov 12, 2023 at 12:36
  • $\begingroup$ Why can't I change x<0 to x<=0 to draw a function graph? Clear["Global`*"] f[x_] := x (x - 1); /; 0 < x <= 1; f[x_] := 2 f[x - 1] /; x >= 1; f[x_] := 1/2 f[x + 1] /; x <= 0; Plot[f[x], {x, -3, 3}] The prompt message is:TerminatedEvaluation["RecursionLimit"] $\endgroup$
    – csn899
    Nov 13, 2023 at 3:30
7
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Plot[With[{qr = QuotientRemainder[x, 1]}, 
  2^First[qr] Last[qr] (Last[qr] - 1)], {x, 0, 3}]
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6
  • 1
    $\begingroup$ Also, Plot[2^IntegerPart[x] FractionalPart[x] (FractionalPart[x] - 1), {x, 0, 3}] $\endgroup$
    – Goofy
    Nov 11, 2023 at 2:00
  • $\begingroup$ If for any x belongs to (-[Infinity], m], there is f [x]>=-8/9, find the value range of m $\endgroup$
    – csn899
    Nov 11, 2023 at 2:12
  • $\begingroup$ I don't think you can say $\forall x \in (-\infty,m] .\exists x . \ \text{such that}\ f(x) \ge -8/9$. That you wrote "there is f[x] >= -8/9" does not change the fact that $x$ is already quantified by "for any x." Did you mean "Find m such that for all x in (-infinity, m], f(x) >= -8/9"? $\endgroup$
    – Goofy
    Nov 11, 2023 at 2:25
  • $\begingroup$ It means that for all x within this interval, its corresponding function values are greater than or equal to -8/9 $\endgroup$
    – csn899
    Nov 11, 2023 at 2:29
  • 1
    $\begingroup$ Perhaps First@Solve[2^IntegerPart[x] FractionalPart[x] (FractionalPart[x] - 1) == -8/9 && 0 <= x <= 3, x, Reals] $\endgroup$
    – Goofy
    Nov 11, 2023 at 2:42
6
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$Version

(* "13.3.1 for Mac OS X ARM (64-bit) (July 24, 2023)" *)

Clear["Global`*"]

f[x_?NumericQ] := x (x - 1); /; 0 < x <= 1;
f[x_?NumericQ] := 2 f[x - 1] /; x >= 1;

xVal = Table[
  x /. FindRoot[f[x] == -8/9, {x, #}][[1]] & /@
   {n + 0.25, n + 0.75},
  {n, {2, 3}}]

(* {{2.33333, 2.66667}, {3.12732, 3.87268}} *)

f2[x_] = 
 ConditionalExpression[f[x], Or @@ (#[[1]] <= x <= #[[2]] & /@ xVal)]

(* ConditionalExpression[f[x], 
 2.33333 <= x <= 2.66667 || 3.12732 <= x <= 3.87268] *)

Plot[{f[x], f2[x]}, {x, 0, 4},
 Filling -> 2 -> Axis,
 AxesStyle -> Arrowheads[0.04],
 GridLines -> {None, {-8/9}},
 GridLinesStyle -> Red]

enter image description here

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1
  • $\begingroup$ How can I draw an image when x<0? $\endgroup$
    – csn899
    Nov 12, 2023 at 12:36

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