Mathematics

Question

What is the y-intercept of the exponential function? f(x)=−32(2)x−3+3

2 Answer

  • I assume your equation is
    [tex]f(x)=-32(2)^{x-3}+3[/tex]

    x intercept is where the line crosses the x axis or where y=0
    set y=f(x)=0 and solve

    [tex]0=-32(2)^{x-3}+3[/tex]
    minus 3 both sides
    [tex]-3=-32(2)^{x-3}[/tex]
    divide both sides by -32
    [tex] \frac{3}{32} =2^{x-3}[/tex]
    take the ln of both sides
    [tex] ln(\frac{3}{32}) =(x-3)ln(2)[/tex]
    [tex] ln(\frac{3}{32}) =xln(2)-3ln(2)[/tex]
    add 3ln(2) to both sides
    [tex] ln(\frac{3}{32})+3ln(2) =xln(2)[/tex]
    divide both sides by ln(2)
    [tex] \frac{ln(\frac{3}{32})+3ln(2)}{ln(2)} =x[/tex]
    [tex] \frac{ln(\frac{3}{4})}{ln(2)} =x[/tex]

    the x intercept is at [tex] x= \frac{ln(\frac{3}{32})+3ln(2)}{ln(2)} [/tex]
    or aprox at x=-0.415037
  • We want to get the y-intercept of the given function, we will see that it is y = 7.

    So we have the exponential function:

    [tex]y = f(x) = 32*(2)^{x - 3} + 3[/tex]

    We want to get the y-intercept, it is just given by evaluating the function in x = 0, by doing that, we will get:

    [tex]y = f(0) = 32*(2)^{0 - 3} + 3 = 7[/tex]

    So the y-intercept of the given function is y = 7.

    If you want to learn more about function's intercepts, you can read:

    https://brainly.com/question/1354826

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