Question

If the function f : R → R be defined by f(x) = 2x – 3 and g : R → R by g(x) = x³ + 5, then find fog and show that fog is invertible. Also, find (fog)⁻¹, hence find (fog)⁻¹(9).

If A = $\left(\begin{array}{l l l }2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 0\end{array}\right)$ , then find value of A² - 3A + 2I.

Show that the relation S in the set R of real numbers defined as S = {(a, b) : a, b ∈ R and a ≤ b³} is neither reflexive nor symmetric nor transitive.

Find the equations of the tangent and the normal, to the curve 16x² + 9y² = 145 at the point $(x_{1},\;y_{1}),$ where x₁ = 2 and y₁ > 0.

Using differentials, find the approximate value of $(3.968)^{1/2}$

Find the equations of the normal to the curve x² = 4y which passes through the point (1, 2).

Consider f :$R_{+}\to[4,\infty)$ given by f(x) = x² + 4 Show that f is invertible with the inverse $f^{-1}$ of f given by $f^{-1}(y)$ = $\sqrt{y-4}$ , where $\textstyle R_{+}$ is the set of all nonnegative real numbers.