Is This Function Continuous? Why is My Calculator Inconsistent?

Oct 2013
713
91
New York, USA
I watched a video showing the six solutions to:

(x^2 - 7x + 11)^(x^2 -13x + 42) = 1

Several websites only found four of the solutions, and even made incorrect graphs. There are three cases that produces two solutions each. The case where the base = 1 and the case where the exponent = 0 were known by all websites, but several did not have the case where the base = -1 and the exponent is even. The solutions are 2, 3, 4, 5, 6, and 7. Desmos made a graph showing (2, 1), (5, 1), (6, 1), and (7, 1). For most of the values of x in between 2 and 5, the value of y doesn't fit on the graph. Using my calculator, x = 1.9 makes y = about 283.23, and x = 2.1 makes y = about .0014. The fact that y changes so much as x ranges from 1.9 to 2.1 makes me wonder if the function is continuous. Does it have holes and/or asymptotes? What real numbers if any are not in the domain? Then I tried x = 2.5, x = 2.9, and x = 3.1. I used my calculator to make L1 be the original, L2 be the base, L3 be the exponent, and L4 be L2^L3. For those three values of x, my calculator says it is a nonreal answer. However, if I take the value in L2 to the exponent L3, it gives an answer. For example, when x = 2.9, the function is -.89^12.71, which my calculator says is -11.3119, but as part of a list it says the answer isn't real. Why?
 

romsek

Math Team
Sep 2015
2,958
1,673
USA
$x^2 - 7x + 11 = \left(x - \dfrac 7 2\right)^2 - \dfrac 5 4$

so you can see there is an interval over which it is negative.

Calculators generally do powers by using logs and taking the log of a negative number is undefined on the reals.
 
  • Like
Reactions: EvanJ

skipjack

Forum Staff
Dec 2006
21,478
2,470
Hence (-.89)^12.71 is likely to be considered non-real by the calculator.