given:
f(x)=x^2
g(x)=2^x
find:
g^-1(f(3))to the nearest tenth
can any one please explain to me how to find the inverse of g of f of 3????
g(x) = 2^x
let y = 2^x
then the inverse function is
x = 2^y
or
y = ln x/ln 2
so g-1(x) = ln x/ ln 2
but f(3) = 3^2 = 9
so g-1(f(3) = ln 9/ln 2 = 3.2
what is "ln x/ln 2"???????
To find the inverse of g(x), we first need to find the inverse function of g(x)=2^x.
1. Start by replacing g(x) with y. So, we have y = 2^x.
2. Swap the x and y variables to get x = 2^y.
3. Solve the equation for y by taking the logarithm base 2 of both sides:
log2(x) = log2(2^y)
log2(x) = y
Therefore, the inverse function of g(x) is g^-1(x) = log2(x).
Next, we want to find the value of g^-1(f(3)):
1. Replace f(x) with its value given in the question: f(3) = 3^2 = 9.
2. Substitute this value into g^-1(x):
g^-1(9) = log2(9)
3. Use a calculator or a logarithm table to calculate log2(9). Using a calculator, we find that log2(9) is approximately 3.1699.
Finally, round the value of log2(9) to the nearest tenth to get the final answer:
g^-1(f(3)) ≈ 3.2