Define function.

Distinguish relation and function with example. Find the domain and range of \(\mathrm f(\mathrm x)\;=\;\sqrt{\mathrm x^2\;–\;2\mathrm x\;–\;8}\;,\;\mathrm x\;\in\;\mathbb{R}.\)

Ans: Domain = (-∞, -2) ∪ [4, ∞); Range = [0, ∞)

Define domain and range of a function.

Find the domain and range of \(\mathrm f(\mathrm x)\;=\;\sqrt{21-4\mathrm x-\mathrm x^2}\;\)

Ans: Domain = [-7, 3], Range = [0, 5]

Define one-one function and onto function.

Let f(x) = x³ + 5, x ∈ R. Find a formula that defines inverse function f ^-1.

\(\mathrm{Ans}:\;\sqrt[3]{\mathrm x-5}\)

Find the domain and range of the function:

a. f(x) = 5 – (x + 3)²

b. f(x) = x / |x|

Ans: (a) Domain = (-∞, ∞), Range = (- ∞, 5]; (b) Domain = R - {0}, Range = {-1, 1}

Show that f(x) = 2x + 3 is bijective (f: R → R). Also find f ^-1(2).

Ans: -1/2

Find the domain and range of the function \(\mathrm f(\mathrm x)=\sqrt{2-\mathrm x-\mathrm x^2}\)

Ans: Domain = [-2, 1]; Range = [0, 3/2]

Show that f: R → R defined by f(x) = cx + d where c (c ≠ 0) and d are real numbers is one to one and onto. Also show that f_of ^-1(x) = x. find f ^-1.

\(\mathrm{Ans}:\;\mathrm f^{-1}(\mathrm x)\;=\;\frac{\mathrm x-\mathrm d}{\mathrm c}\)

Let f: R → R and g: R → R be defined by f(x) = x³ + 2 and g(x) = 4x – 1. Find (f_og) (x) and (g_of)(x). Is (f_og)(x) = (g_of)(x)? Are (f_og) (x) and (g_of) (x) one to one?

Ans: (4x - 1)³ + 2, 4x³ + 7, No, Yes, Yes

Define composite functions of two functions f and g.

Let f: R → R and g: R → R be defined by f(x) = 3x² - 4 and g(x) = 2x - 5, find (f_og) (x) and (g_of) (x). Are the functions (f_og) (x) and (g_of) (x) one to one? Give reasons.

Ans: 12x² - 60x + 71, 6x² - 13, No

Let the function f(x) = x³ and g(x) = sinx, x ∈ R. Find f_og and g_of. Is f_og = g_of? Examine whether f is one to one and onto or not.

Ans: f_og(x) = sin3x; g_of(x) = sin x³ ; No; One to One and Onto

Define one-one onto and one-one into function. Show that the function f: [1 ,4] → R defined by f(x) = x² is one-one but not onto.