+44 1.) Let $f(x)=\left\lfloor\left(-\frac58\right)^x\right\rfloor$ be a function that is defined for all values of $x$ in $[0,\infty)$ such that $f(x)$ is a real number. How many distinct values exist in the range of $f(x)$?
2.) Let \[f(x) = \left\{\begin{array}{cl}ax+3 & \text{ if }x>0, \\ab & \text{ if }x=0, \\bx+c & \text{ if }x<0.\end{array}\right.\]If $f(2)=5$, $f(0)=5$, and $f(-2)=-10$, and $a$, $b$, and $c$ are nonnegative integers, then what is $a+b+c$?
WITH LATEX:
1.) Let \(f(x)=\left\lfloor\left(-\frac58\right)^x\right\rfloor\) be a function that is defined for all values of \(x\) in \([0,\infty)\) such that \(f(x)\) is a real number. How many distinct values exist in the range of \(f(x)\)?
2.) Let
\(f(x) = \left\{\begin{array}{cl}ax+3 & \text{ if }x>0, \\ab & \text{ if }x=0, \\bx+c & \text{ if }x<0.\end{array}\right.\)
If \(f(2)=5\), \(f(0)=5\), and \(f(-2)=-10\), and \(a\), \(b\), and \(c\) are nonnegative integers, then what is \(a+b+c\)?
+118609 2)
f(2)=2a+3=5
2a=2
a=1
f(0)=ab=5
1*b=5
b=5
f(-2)=b*-2+c=-10
-2b+c=-10
-2*5+c=-10
-10+c=-10
c=0
a+b+c = well you can add up single digit numbers I expect.
+118609 1.) Let \(f(x)=\left\lfloor\left(-\frac58\right)^x\right\rfloor\) be a function that is defined for all values of x in \([0,\infty)\) such that f(x) is a real number. How many distinct values exist in the range of f(x)?
The real solutions (-5/8)^x are always going to be between -1 and 1 not inclusive (when x is positive)
so the floor function can only be -2, -1 and 0
Hence you can answer the actual question.
