Make a conjecture about the equation of the following exponential functions, named f.
Suppose that:
f\left(x\right) = a^x
Since the function is decreasing and the graph is above the x -axis, we deduce that:
0 \lt a \lt 1
Furthermore, we can see on the graph that:
f\left(2\right)=\dfrac{4}{9}
So, we can write:
f\left(2\right) = \dfrac{4}{9}
a^2 = \dfrac{4}{9}
As a>0, we get:
a=\dfrac{2}{3}
This could be the graph of f : x \longmapsto \left( \dfrac23 \right)^x.
Suppose that:
f\left(x\right) = a^x
Since the function is increasing and the graph is above the x -axis, we deduce that:
a \gt 1
Furthermore, we can see on the graph that:
f\left(-1\right)=\dfrac{2}{3}
So, we can write:
f\left(-1\right) = \dfrac{2}{3} \\a^{-1} = \dfrac{2}{3} \\a=\dfrac{3}{2}
This could be the graph of f : x \longmapsto \left( \dfrac32 \right)^x.
Since the graph is above the horizontal asymptote which is y=1, we can derive:
f\left(x\right) =1+ a^x
Since the function is decreasing, we deduce that:
a \gt 1
The only acceptable answer is :
f : x \longmapsto 1+2^x
This could be the graph of f : x \longmapsto 1+2^x.
Since the graph is below the horizontal asymptote which is the x-axis, we get:
f\left(x\right) = -a^x
Since f is decreasing, then -f is increasing. Notice that:
-f\left(x\right) = a^x
So, we must have:
a \gt 1
Therefore the only acceptable answer is:
f\left(x\right)=-2^x
This could be the graph of f : x \longmapsto -2^x.
Since the graph is below the horizontal asymptote which is y=1, we can deduce that:
f\left(x\right) = -a^x +1
As f is decreasing, we conclude that the graph of:
y=a^x
is increasing. Therefore:
a \gt 1
On the other hand, we can see on the graph that:
f\left(1\right)=-2
Therefore, the only acceptable answer is:
f\left(x\right) = -3^x+1 = 1-3^x
This could be the graph of f : x \longmapsto 1-3^x.
Suppose that:
f\left(x\right) = a^x
Since the function is increasing and the graph is above the x -axis, we deduce that:
a \gt 1
Furthermore, we can see on the graph that:
f\left(1\right)=\dfrac{4}{3}
a = \dfrac{4}{3}
Therefore:
f\left(x\right)=\left(\dfrac{4}{3}\right)^x = \left(\dfrac{3}{4}\right)^{-x}
This could be the graph of f : x \longmapsto \left( \dfrac34 \right)^{-x}.
Since the graph is below the horizontal asymptote which is the x-axis, we get:
f\left(x\right) = -a^x
Since f is increasing, then -f is decreasing. Notice that:
-f\left(x\right) = a^x
So, we must have:
0 \lt a \lt 1
Therefore, the only acceptable answer is:
f\left(x\right)=-\left(\dfrac{1}{2}\right)^x
This could be the graph of f : x \longmapsto -\left( \dfrac12 \right)^x.