Before we can find the inverse of the given function, we need to rewrite it as an equation relating x and y.
f(x)=sqrt(4x-1) ⇔ y=sqrt(4x-1)
Now, to algebraically determine the inverse of the given equation we exchange x and y and solve for y.
Given Equation & Inverse Equation
y=sqrt(4 x-1) & x=sqrt(4 y-1)
The result of isolating y in the new equation will be the inverse of the given function.
Now that we have found y, we know the inverse of the given function.
f^(- 1)(x)=x^3+1/4
b
Similarly as in Part A, to find the inverse of a function we start by exchanging the variables in the given function. Then we can solve for y.
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Given Function & & Inverse Function
y=log_7 x & & x=log_7 yNow we have to isolate y in the inverse equation. The inverse of a logarithmic function is an exponential function. Therefore, we can use the definition of a logarithm to rewrite the equation.
Definition:& x=log_b y &&⇔ b^x= y
Equation:& x=log_7 y &&⇔ 7^x= y
We found that y=7^x. This is the inverse function.
g^(-1)(x) = 7^x
