Given the functions
f(x)=x+5/3 and g(x)=1/f^-1(x)+1, find the value of g(3).
The first step would first be to find the inverse of x+1, the denominator of the fraction. I think the inverse would be 1-x. And now we have 1/1-x so we can just plu in 3. Right?
I have a feeling you meant f(x) = (x+5)/3
or is it the way you typed it ?
confirm please
Yes it is (x+5)/3 thank you for correcting me.
Given the functions
f(x)=(x+5)/3 and g(x)=1/f^-1(x)+1, find the value of g(3).
The first step would first be to find the inverse of x+1, the denominator of the fraction. I think the inverse would be 1-x. And now we have 1/1-x so we can just plu in 3. Right?
f(x) = (x+5)/3
f^-1(x) = 3x-5
g(x) = 1/(f^-1(x)+1) = 1/(3x-4)
g(3) = 1/5
Of course, you might also have meant just what you typed, in which case
g(x) = 1/f^-1(x) + 1 = 1/(3x-5) + 1 = (3x-4)/(3x-5)
and so g(3) would be 5/4
To find the value of g(3), we need to first find the inverse of the function f(x) = x + 5/3.
To find the inverse of a function, you need to swap the roles of x and y and solve for y.
So, let's start by swapping x and y in the function f(x):
x = y + 5/3
Now, let's solve for y:
y = x - 5/3
So, the inverse function of f(x) = x + 5/3 is f^(-1)(x) = x - 5/3.
Now that we have the inverse function, we can find the value of g(3) by substituting x = 3 into the function g(x) = 1/f^(-1)(x) + 1.
g(3) = 1/f^(-1)(3) + 1
Substituting x = 3 into the inverse function f^(-1)(x):
g(3) = 1/(3 - 5/3) + 1
Now, let's simplify:
g(3) = 1/(9/3 - 5/3) + 1
g(3) = 1/(4/3) + 1
To divide by a fraction, we can multiply by its reciprocal:
g(3) = 1 * (3/4) + 1
g(3) = 3/4 + 1
Now, let's find a common denominator and add the fractions:
g(3) = (3/4 + 4/4)
g(3) = 7/4
Therefore, the value of g(3) is 7/4.