Question

Determine whether or not the given function is one-to-one and, if so, find the inverse. If f(x)=(−4x+7)⁴ has an inverse, give the domain of f−¹.

a) f−¹(x)=(7−x(¹/⁴))/4; domain: (0,[infinity])
b) f−¹(x)=(7+4x)(¹/⁴); domain: (−[infinity],7/4)
c) Not one-to-one
d) f−¹(x)=(x(¹/⁴)−7)/4; domain: (−[infinity],[infinity])
e) f−¹(x)=(7−x(¹/⁴))/4; domain: (7/4,[infinity])

Answer 1

To determine if a given function is one-to-one, we check if different inputs produce different outputs. The inverse function can be found by solving f(x) = y for x.

Step-by-step explanation:

To determine whether the given function is one-to-one, we need to check if two different inputs produce different outputs. Let's check:

f(x) = (-4x+7)^4

Suppose we have two inputs x1 and x2 such that f(x1) = f(x2).

Then, (-4x1+7)^4 = (-4x2+7)^4. Solving this equation for x1 and x2 will help us determine if the function is one-to-one.

Now, to find the inverse function, we need to find an expression for x in terms of f(x). Let's solve f(x) = y for x.

y = (-4x+7)^4

Taking the fourth root of both sides, we get (y)^(1/4) = -4x + 7

Rearranging the equation, we get x = (7-y^(1/4))/4, which is the inverse function.