Important Questions - Answers
1. If the function f is defined by f(x) = x + 2, x > 1
= 2, -1 ≤ x ≤ 1
= x -1, -3 < x < 1
find (i) f (3) (ii) f(0) (iii) f(-1.5) (iv) f(2) + f(-2) (v) f(-5)
Sol: (i) f(x) = x + 2, x > 1
f(3) = 3 + 2 = 5
(ii) f(x) = 2, - 1 ≤ x ≤ 1
f(0) = 2
(iii) f(x) = x - 1, -3 < x < 1
f(-1.5) = -1.5 - 1 = -2.5
(iv) f(x) = x + 2
f(x) = 2 + 2 = 4
f(x) = x - 1
f(-2) = -2 - 1 = -3
f(2) + f(-2) = 4 - 3 = 1
v) f(-5) cannot be found out by using the given definition of f(x).
2. If f: R R is defined by f(x) = , then show that f (tan ) = cos 2
Sol:
3. If f : R - {±1} R is defined by f(x) = log ,
then show that = 2f(x).
Sol:
4. If A = {1, 2, 3, 4} and f : A R is defined by f(x) = , find the range of f.
Sol:
5. Verify if f : R (0, ) defined by f(x) = 2x is a bijection or not?
Sol:
6. Is g = {(1, 1), (2, 3), (3, 5), (4, 7)} a function from A to B, where A = { 1, 2, 3, 4} and B =
{1, 3, 5, 7}. If g(x) is defined as g(x) = ax + b, then find 'a' and 'b'.
Sol: A = {1, 2, 3, 4}; B = {1, 3, 5, 7}
g = {(1, 1), (2, 3), (3, 5), (4, 7)}
g (1) = 1, g(2) = 3, g(3) = 5, g(4) = 7
This suits the definition of a function
f is a function from A to B
g(x) = ax + b, x A
g(1) = a(1) + b = a + b a + b = 1 (1)
g(2) = a(2) + b = 2a + b 2a + b = 3 (2)
g(3) = a(3) + b = 3a + b 3a + b = 5 (3)
g(4) = a(4) + b = 4a + b 4a + b = 7 (4)
Solving any two equations, we get a = 2, b = -1.
7. If f(x) = ex and g(x)= logex, then show that gof= fog and find f-1 and g-1.
Sol: f(x) = ex, g(x) = logex
(gof)(x) = g[f(x)]
= g[ex]
= loge ex
= x (logee)
= x (1)
(fog)(x) = f[g(x)]
= f(logex)
= elogex
= x (2)
(1), (2) (gof)(x) = (fog)(x) gof = fog
Let y = f(x) = ex
f-1(y) = x; y = ex
logey = x
x = x
f-1 (y) = logey
f-1 (x) = logex (3)
Let z = g(x) = logex
g-1 (z) = x, z = logex
ez = x
x = x
g-1(z) = ez
g-1 (x) = ex
(1), (2), (3), (4) The result.