Respuesta :
A power function with two points can be written by substituting the values of the two points in the given function.
To given an example, let f(x) = aˣ b;
Taking natural logs on both sides:
ln f(x) = ln (aˣb);
ln f(x) = ln (aˣ) + ln (b);
ln f(x) = xln (a) + ln (b);
Taking Exponential both sides
then,
Let the two points be P (x₁, y₁) and Q (x₂, y₂)
The we get two equations:
(i) ln f(y₁) = x₁ ln (a) + ln (b);
(ii) ln f(y₂) = x₂ ln (a) + ln (b);
Subtracting (ii) from (i)
ln f(y₁) - ln f(y₂) = x₁ ln (a) - x₂ ln (a) + ln (b) - ln (b);
ln f(y₁)/f(y₂) = ln a( x₁ - x₂)
[ln f(y₁)/f(y₂)]/ [( x₁ - x₂)] = ln a
Taking exponential both side
let f(y₁)/f(y₂) = s
let ( x₁ - x₂)= r
[tex]e^{ \frac{ln s}{r} } = e^{ ln a}[/tex]
[tex]e^{ \frac{ln s}{r}[/tex] = a
On solving both the equations we get two values of b
f(y₁) = aˣ₁ . b
f(y₂) = aˣ₂ . b
ln f(y₁) = ln (aˣ₁) + ln (b); --(1)
ln f(y₂) = ln (aˣ₂) + ln (b); --(2)
subtracting (2) from (1)
ln f(y₁) - ln(y₂) = ln([tex]e^{\frac{ln s}{r}^x1}[/tex]) - ln([tex]e^{\frac{ln s}{r}^x2}[/tex]) + ln(b) - ln(b)
ln f(y₁)/f(y₂) = (ln s/r)ˣ₁ - (ln s/r)ˣ₂
ln f(y₁)/f(y₂) = x₁ ln s/rˣ₁ - x₂ ln s/rˣ₂
rˣ₁rˣ₂ ln f(y₁)/f(y₂) = x₁ rˣ₂ ln s - x₂ rˣ₁ ln s
ln [tex]f(y1)/f(y2)^{r^{x1} r^{x2}}[/tex] = ln [tex]s^{x1 r^{x2}}[/tex] - ln [tex]s^{x2 r^{x1}}[/tex]
ln [tex]f(y1)/f(y2)^{r^{x1} r^{x2}}[/tex] = ln [tex]\frac {s^{x1 r^{x2}}}{s^{x2 r^{x1}}}[/tex]
Taking exponential both sides.
let [tex]f(y1)/f(y2)^{r^{x1} r^{x2}}[/tex] = w
let [tex]\frac {s^{x1 r^{x2}}}{s^{x2 r^{x1}}}[/tex] t
then,
[tex]e^{ln w} } = e^{ ln t}[/tex]
w = t
[tex]f(y1)/f(y2)^{r^{x1} r^{x2}}[/tex] = [tex]\frac {s^{x1 r^{x2}}}{s^{x2 r^{x1}}}[/tex]
Solving this equation we get the value of x by putting the values given
Learn more about Power Functions at:
brainly.com/question/29546964
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