I'm struggling with graphs. Any help appreciated. Thanks
The figure shows the graphs off(x) = x3 and g(x) = ax3. What can
you conclude about the value of a?
a. a < –1
b . –1 < a < 0
c . 0 < a < 1
d . 1 < a
I'm struggling with graphs. Any help appreciated. Thanks
The figure shows the graphs off(x) = x3 and g(x) = ax3. What can
you conclude about the value of a?
a. a < –1
b . –1 < a < 0
c . 0 < a < 1
d . 1 < a
We see the following:
1)$\displaystyle f(x) = g(x) \mbox{ for }x=0$
2)$\displaystyle f(x) > g(x) \mbox{ for }x>0$
3)$\displaystyle f(x) < g(x) \mbox{ for }x<0$
Let us examine #2:
We know that $\displaystyle f(x) = x^3 \mbox{ and }g(x)=ax^3$
Thus,
$\displaystyle x^3 > ax^3 \mbox{ for }x>0$
$\displaystyle x^3 - ax^3>0$
$\displaystyle x^3 (1 - a) > 0$
Since $\displaystyle x>0 \Rightarrow x^3 >0$
Thus, $\displaystyle (1-a)>0 \Rightarrow a<1$.
Let us examine #3:
We know that $\displaystyle x^3 < ax^3 \mbox{ for } x<0$
$\displaystyle x^3 - ax^3 < 0$
$\displaystyle x^3 ( 1 - a) < 0$
This leads to, $\displaystyle a<1$ but we already know that, so we did not gain anything.
But if we look at $\displaystyle g(x)$ we see that $\displaystyle g(x)>0$ for $\displaystyle x>0$.
Thus,
$\displaystyle ax^3 \mbox{ for }x>0$
Since $\displaystyle x>0$ it tells us that $\displaystyle a>0$.
Thus,
$\displaystyle 0<a<1$.