Answer:
f^-1(x) = ± (x/4)^¼
f^-1(x) is not a function
Option (B ), y = ± (x/4)^¼ , f^-1(x) is not a function is correct option
step-by-step explanation :
★ To find the inverse of function , solve y :
x = f(y)
x = 4y⁴
x/4 = y⁴
→f^-1(x) = ± ⁴√x/4
f(x) is an even function, so f(-x) = f(x). Then the inverse relation is double-valued: for any given y, there can be either of two x-values that will give that result.
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A function is single-valued. That means any given domain value maps to exactly one range value. The test of this is the "vertical line test." If a vertical line intersects the graph in more than one point, then that x-value maps to more than one y-value.
The horizontal line test is similar. It is used to determine whether a function has an inverse function. If a horizontal line intersects the graph in more than one place, the inverse relation is not a function.
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Since the inverse relation for the given f(x) maps every x to two y-values, it is not a function. You can also tell this by the fact that f(x) is an even function, so does not pass the horizontal line test. When f(x) doesn't pass the horizontal line test, f^-1(x) cannot pass the vertical line test.
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The attached graph shows the inverse relation (called f₁(x)). It also shows a vertical line intersecting that graph in more than one place.