Question

Let f:R→R be a function defined by f(x) = 5x³ - 8 for all x ∈ R, show that f is one-one and onto. Hence find f⁻¹.

Answer 1

it is given that f : R ----> R be a function defined by f(x) = 5x³ - 8 for all x belongs to real numbers .

let's take $x_1$ and $x_2$ in domain of given function f(x) such that,
$f(x_1)=f(x_2)$

or, $5x_1^3-8=5x_2^3-8$

or, $5x_1^3=5x_2^3$

or, $x_1=x_2$

hence it is clear that, f is one one function.

given function, f(x) = 5x³ - 5 is a polynomial function. we know, every polynomial function is defined in all real value of x and range of polynomial function belongs to all real numbers.

e.g., range = co - domain $\in$ R

so, f is an onto function.

now, f(x) = y = 5x³ - 8

or, y + 8 = 5x³

or, (y + 8)/5 = x³

or, x = ³√{(y + 8)/5}

hence, $f^{-1}=\sqrt[3]{\frac{(x+8)}{5}}$