a) Show that $3x+7$ is $\Theta(x)$.
b) Show that $2x^2 +x -7$ is $\Theta(x^2)$
$2x^2+x-7 \geq x^2$ for $x \geq 2$
if $x \gt 1$
$x^2 \gt x$
$2x^2 \gt 2x$
$x^2 \gt 1$
$x^2\geq x^2$
c) Show that $\lfloor x+.5\rfloor$ is $\Theta(x)$
d) Show that $\log_{10}(x)$ is $\Theta(\log_2(x))$
I'm not very sure about if I'm setting these up the right way and I'm not sure how to start a, c, and d.
Here's a sample. To show $3x+7\in \Theta(x)$ we need to show two things:
To show (1) we need a $c>0$ and $k$ such that $3x+7<c\,x$ for all $x>k$. Suppose we pick $c=4$, then we need to find when $3x+7<4x$, but that holds whenever $7<x$ so we can use $k=7$.
To show (2) we need a $d>0$ and $k$ such that $3x+7>d\,x$ for all $x>k$. The obvious choice is to pick $d=3$, since it's clear that $3x+7>3x$, for all $x$, so we can use $k=0$ and conclude that $3x+7 > 3x$ for all $x>0$ and we're done.