Let f be a one-to-one function whose inverse function is given by the formula
f^(-4)(x)=x^5+2x^3x+1
a. Computer f^-1(1) and f(1)
b. computer the value of x such that f(x)=1
C. Computer the value of y such that f^-1(y)=1
Please help me solve this problem.
i made error in typing the function.
it is
f^-4(x)=x^5+2x^3+3x+1
I have heard of f^(-1)(x) as the inverse of f(x) but I have no idea what wou mean by f^(-4)(x)
oh yeah that was an error. sorry about that.
it is f^(-1)(x)=x^5+2x^3+3x+1
Sure, let's solve the problem step by step.
a. To compute f^(-1)(1), we can use the formula for the inverse function f^(-1)(x):
f^(-1)(x) = x^5 + 2x^3 + 1
Substituting x = 1 into the formula, we get:
f^(-1)(1) = 1^5 + 2(1)^3 + 1
= 1 + 2 + 1
= 4
Therefore, f^(-1)(1) = 4.
To compute f(1), we can use the original function f(x):
f(x) = x^5 + 2x^3 + 1
Substituting x = 1 into the formula, we get:
f(1) = 1^5 + 2(1)^3 + 1
= 1 + 2 + 1
= 4
Therefore, f(1) = 4.
b. To find the value of x such that f(x) = 1, we need to solve the equation:
f(x) = 1
Using the original function f(x), we get:
x^5 + 2x^3 + 1 = 1
Simplifying the equation, we have:
x^5 + 2x^3 = 0
Factoring out x^3, we have:
x^3(x^2 + 2) = 0
Therefore, the equation is satisfied when either x^3 = 0 or x^2 + 2 = 0.
From x^3 = 0, we find:
x = 0
From x^2 + 2 = 0, we find:
x^2 = -2
Taking the square root of both sides, we get:
x = ±√(-2)
Therefore, the values of x such that f(x) = 1 are x = 0 and x = ±√(-2).
c. To find the value of y such that f^(-1)(y) = 1, we can use the formula for the inverse function:
f^(-1)(x) = x^5 + 2x^3 + 1
Substituting y = 1 into the formula, we get:
f^(-1)(1) = 1^5 + 2(1)^3 + 1
= 1 + 2 + 1
= 4
Therefore, the value of y such that f^(-1)(y) = 1 is y = 4.
I hope this helps! Let me know if you have any further questions.