The figures below show the graphs of the exponential functio

The figures below show the graphs of the exponential functions f(x) and g(x), and the linear function, h(x). The function f(x) has y-intercept 0.75 and goes through the point (1, 6). The function g(x) has y-intercept 2 and goes through the point (2, 2/9). The function h(x) has y-intercept 3 and goes through the point (alpha, alpha + 3). Find a formula for f(x) = ______help (formulas) Find a formula for g(x) = _________help (formulas) Find a formula for h(x) = _______help (formulas) Find the exact value(s) of x such that f(x) = g(x). If there is more than one solution, enter your answers as a comma separated list. x = _____________help (numbers)

Solution

(a)

f(x) show exponential growth.             // increase in value of \"x\" value of \"y\" also increases.

so the general formula of ecponential growth is y = a*e^x

this graph passes through (0, 0.75)

plugging these values in the equation, we get

0.75 = a*e⁰                                    // e⁰ = 1

0.75 = a * 1

a = 0.75

so the formula is y = 0.75 e^x

(b)

since this is exponential decay

the general equation of this graph is y = a * e^-x.

this graph passes through (2, 2/9)

2/9 = a * e^-2

a *0.135 = 2/9

a = 1.65

so the equation if the graph is y = 1.65 e^-x

(c)

this is a line.

it has y-intercept \"3\", so it passes through (0, 3)

and this line passes through (a, a+3)

we can find the slope of this line as we have two points

slope (m) = (a+3 - 3) / (a - 0)             // m = (y₂ - y₁) / (x₂ - x₁)

m = a / a = 1

m = 1

now we have the slope and one point (0,3)

using point slope form of a line => y - y₁ = m (x - x₁)

y - 3 = 1 (x - 0)

y - 3 = x

x - y + 3 = 0

(d)

f(x) = g(x)

0.75 e^x = 1.65 e^-x

0.75 e^x = 1.65 * (1/e^x )                     // x^-a = 1 / x^a

e^x * e^x = 1.65 / 0.75

e^2x = 2.2                         // x^a * x^a = x^{a + a} = x^2a

taking natural log both sides

ln e^2x = ln 2.2

2x ln e = ln 2.2                // ln a^b = b* ln a

2x = 0.7884                  // ln e = 1

x = 0.394