Question

The figures below show the graphs of the exponential functions and $g(x)\mid$ , and the linear function, $h(x)\mid$ . The function $f(x)\mid$ has y-intercept 0.75 and goes through the point (1,3). The function $g(x)\mid$ has y-intercept 2 and goes through the point (2,2/9). The function $h(x)\mid$ has y-intercept 5 and goes through the point (a,a+5).

f(a)

(a) Find a formula for $f(x)=\mid$

(b) Find a formula for $g(x)=\mid$

(c) Find a formula for $h(x)=\mid$

(d) Find the exact value(s) of $x=\mid$ such that $f (x) = g (x)\mid$

If there is more than one solution, enter your answers as a comma separated list.

$x=\mid$

Answer 1

For the first two, we use f(x) = a b^x

For the first graph , points are (0, 0.75) , (1,3)

That gives

3= 0.75(b)¹

b= 3/ 0.75 = 4

Equation is y= 0.75(4)^x

For the second graph, points are (0,2),(2, 2/9)

2/9 = 2(b)²

1/ 9 = b²

b= 1/3

Equation is y = 2(1/3)^x

For the third equation , points are (0,5),(a,a+5)

Slope = (y₂-y₁)/(x₂-x₁) = (a+5-5)/(a-0)=1

Equation of line is y= x+5

Part d. f(x)= g(x)

0.75(4)^x=2(1/3)^x

(4/1/3)^x=2/0.75

12^x=8/3

x log 12 = log(8/3)

x =log(8/3)/log(12)