Prove that the real valued function f x = x e x - 1 + x 2 + 1 is an even function on R \ 0 .

The given function is f x = x e x - 1 + x 2 + 1 . . . i . Here, f - x = - x e - x - 1 - x 2 + 1 . . . ii Subtracting ii from i , we have x e x - 1 + x 2 + 1 - - x e - x - 1 - x 2 + 1 = x e x - 1 + x 2 + 1 + x e - x - 1 + x 2 - 1 = x e x - 1 + x 2 + x e - x - 1 + x 2 = x e x - 1 + e x · x 1 - e x + x = x e x - 1 - e x · x e x - 1 + x = x - e x · x e x - 1 + x = x 1 - e x e x - 1 + x = - x e x - 1 e x - 1 + x ∴ f x - f - x = - x + x = 0 Thus, the given function f is an even function.

Find the domain and range of the function f x = tan π x 1 + sin π x + x 2

The given function is f x = tan π x 1 + sin π x + x 2 . Here, x always return values in form of integers only and tan n π = 0 , sin n π = 0 and x 2 is always positive. ⇒ f x = tan π x 1 + sin π x + x 2 = 0 1 + 0 + x 2 = 0 Thus, the domain of f is R and as the given function always return the value 0 for any real number so the range of f is 0 .

Find the domain and range of the function f x = x 2 - 3 x

The given function is f x = x 2 - 3 x . Here, f to be a real valued function 2 - 3 x ≠ 0 ⇒ x ≠ 2 3 ⇒ x ∈ R - 2 3 Let, f x = y ⇒ y = x 2 - 3 x ⇒ 2 y - 3 x y = x ⇒ 2 y = x + 3 x y ⇒ x = 2 y 3 y + 1 Now, 1 + 3 y ≠ 0 ⇒ 3 y ≠ - 1 ⇒ y ≠ - 1 3 ∴ x ∈ R / - 1 3 Thus, the domain and range of the function are R / 2 3 and R / - 1 3 respectively.

E M B I B E

Intermediate First Year Mathematics paper - 1A>Algebra, Vector Algebra, Trigonometry>Chapter 1 - Functions>Exercise 1 (c)

Telangana Board Solutions for Exercise 3: Exercise 1 (c)

Author:Telangana Board

Telangana Board Mathematics Solutions for Exercise - Telangana Board Solutions for Exercise 3: Exercise 1 (c)

Attempt the practice questions from Exercise 3: Exercise 1 (c) with hints and solutions to strengthen your understanding. Intermediate First Year Mathematics paper - 1A solutions are prepared by Experienced Embibe Experts.

Questions from Telangana Board Solutions for Exercise 3: Exercise 1 (c) with Hints & Solutions

MEDIUM

11th Telangana Board

IMPORTANT

Intermediate First Year Mathematics paper - 1A>Chapter 1 - Functions>Exercise 1 (c)>Q II

Find the value of $a$ if the domain of the real valued function $f (x) = \frac{1}{\sqrt[3]{(x - 2)} \log_{(4 - x)} 10}$ is $(- \infty, a) - \{2, 3\}$ .

HARD

11th Telangana Board

IMPORTANT

Intermediate First Year Mathematics paper - 1A>Chapter 1 - Functions>Exercise 1 (c)>Q II

Find the value of $x$ if the domain of the real valued function $f (x) = \sqrt{\frac{4 - x^{2}}{[x] + 2}}$ , where $[x]$ is the greatest integer function is given as $(- \infty, x) \cup [- 1, 2]$ .