EXERCISE-A
1) If f(x)= 2x² - √x +1, find
a) f(4). 31
b) f(0). 1
c) f(1/4). 5/8
2) If f: x --> (x²-4)/(x-2) then find
a) f(1). 3
b) f(2). Undefined
3) If log₂(x² + 3x +4), find
a) f(1). 3
b) f(4). 5
4) If the function f: N --> N is defined by f(x)=√x, then find f(25)/{f(16)+f(1)}. 1
5) If f(x)= x³/3 - x²/2 + x -16, then find f(1/2). -187/12
6) if f(x)= 7x⁴- 2x³- 8x -5, find f(-1). 12
7) If f(x)=2x²- 3 √x +2, find
A) f(0). 1
B) f(4). 27
C) f(h+2). 2h+3 - 3 √(h+1)
8) If f: x-->log(x³+2x+3), then find f(4). 3
9) If f(x)= sin²x + cosx, find
a) f(π/6). 1/4 + √3/2
b) f(3π/6). 1
c) f(-π/4). 1/2 + 1/√2
10) If f(x)= 3 cos²x - 2sinx. Find
a) f(π/3). 3/4 - √3
b) f(4π/3). 3/4 +√3
c) f(-π/6). 13/4
11) Find {f(x+h) - f(x)}/h when
a) f(x)= 4x²+ 2x -3. 2(4x+ 2h+1)
b) f(x)= log x. 1/h log(1+ h/x)
c) f(x)= (1-x)/(1+x). -2/{(1+x)(1+x+h)}
12) f(x)= (x²- 5x+6)/(x²- 8x +12), find
A) f(2). 0
B) f(-5).
C) f(0).
D) f(a).
E) f(h+1).
F) f(h-1)
G) f(2+ h).
H) f(2/h).
13) If f(x)= (ax + b)/(bx + a), find f(x) f(1/x).
14) if g(x)= (x - a)/x + x/(x - b), find the value of g{(a+b)/2}.
15) If f(x)= eˣ. Then find
A) f(x +2).
B) f(x +2)
C) f(x - y)
D) f{(x+1)/(x-1)}
E) f(log₅x).
16) If f(x)= aˣ then find
A) f(x)f(y)
B) f(x-1)f(x+1)
EXERCISE -B
1) 2x -1 when x≤2
If f(x)= x² -1 when 2 < x < 3
2x +2 when x≥ 3
Find
a) f(-1). -3
b) f(2). 1
c) f(2.5). 5.25
d) f(3). 8
e) f(3.5). 8
2) f(x) = 2x-1, when x≤ 0
x², when x > 0.
Find
a) f(1/2). 1/4
b) f(-1/2). -2
3) 1+x, -1≤ x <0
If f(x)= x² - 1, 0< x < 2
2x, 2 ≤ x
Find
a) f(3). 6
b) f(-2). -1
c) f(1/2). 3/4
d) f(2-h) h²- 4h+3
e) f(-1+ h). h
4) 3x-2 when x ≤ 0
If f(x)= x+ 1 when x > 0
Find
A) f(-1). -2
B) f(0). -5
5) 2x²+1 ; x ≤ 2
If f(x)= 1/(x -2) ; 2< x ≤ 3
2x -5 ; x > 3
Then find the value of
A) f(-1)
B) f(0)
C) f(√2)
D) f(-2)
E) f(4)
F) f(2.5)
EXERCISE - C
1) If f(x)= log{(1-x)/(1+x), then f(p) + f(q)= f{(p+q)/(1+pq)}.
2) If f(x)= 2x √(1- x²), then f(sin (x/2))= sin x
3) If f(x)= cos(log x), then f(1/x). f(1/y) - 1/2[f(x/y) + f(xy)]= 0.
4) If f(x)= x(x-a)/(b-a) + x(x-b)/(a-b), then f(a) + f(b)= f(a+b).
5) If f(x)= (x-1)/(x+1), then {f(x) - f(y)}/{1+ f(x). f(y)}= (x-y)/(x+y)
6) If f(x) = log{(1+x)/(1-x)}, then f{(2x/(1+x²)} = 2 f(x).
7) If f(x) = (x-p)/x + x/(x-q) then f{(p+q)/2}= 4pq/(p² - q²).
8) If f(x) = eᵖˣ⁺ᑫ, then f(a). f(b). f(c) = f(a+b+c)e²ᑫ.
9) If If f(x)= tan x then f(x + y)= {f(x) + f(y)}/{1- f(x)f(y)}
10) If g(x)= (1- x)/(1+ x) then g(cos 2x)= tan²x.
11) If f(x)= (ax+b)/(bx+a), then f(x).
f(1/x)= 1.
EXERCISE - D
1) If f(x)= (ax + b)/(bx - a), find f{f(1/x)}. 1/x
2) If f(x)= (1- x)/(1+ x), then show that f{f(x)} = x.
3) f(x)= (x+1)/(x+2), find f{f(1/x)}. (3x+2)/(5x+3)
4) if f(x) = (4x -5)/(3x -4), find the value of f{f(x)}.
5) if f(x)= 1/(1+ x) then find f[f{f(x)}].
EXERCISE - E
1) If f(2x -1)= (3x+1)/(x-1) then find f(2- x). (11-3x)/(1-x)
2) f{(x-1)/(2x+1)= 2- x , then find f(3x-1). (2-5x)/(1-2x)
3) If f(2x-1)= (3x-1)/(x+1) find
a) f{f(4)}. 23/17
b) f{f(1-3x)}. (8-15x)/(8-9x)
4) If g(x -1)= 7x -5 then find the value of
g(x +2).
5) if g(2x -1)= (x+1)/(x+2). Then find
A) g(x).
B) g(2)
C) g(0)
D) g(h+1)
E) g(2+ h).
F) g(2/h)
G) g(- x)
6) Given f{(x-2)/(x+3)= (x -1)/(2x+1), find the value of f(5 -2x)
7) f(x+3)= 3x²- 2x +5, then find the value of f(x -1).
8) If f(x +1)= x³ then find f(1- x).
EXERCISE - F
1) y = f(x)= (x+1)/(x+2). Then find
A) f(y).
B) f(1/x).
C) f{f(x)}.
2) y= f(x) =(ax + b)/(ax - a), then find the value of f(y).
3) If y= f(x)= (2-x)/(5+3x) and z= f(y), express z in terms of x. (7x+8)/ (12x+31)
4) If y= (x -3)/(2x+1) and z= f(y), express z = f(x).
5) If y= f(x)= (5x+3)/(4x-5) then show that f(y)= x
6) If y= f(x)= (3x+1)/(3x-m) and f(y)= x, find m. 2
7) If y= f(x)= (3x+4)/(5x-m) and f(y)= x, find m. 3
8) If y= f(x)= (3x -5)/(2x-m) and f(y)= x, find m.
EXERCISE - G
1) If f(x) = 2x² - 5x +4, for w
hat value of x is 2f(x) = f(2x)
2) Let f(x) =10x²- 13x +13. Find the value of x for which f(x) =16.
3) If f(x) =(a - xⁿ)¹⁾ᵐ , where n is a positive integer. Find the value of f{f(x)}.
4) if f(x) =x + a and g(x) = x - a, find the value of {f(x)}² - {g(x)}².
5) If eʸ + e⁻ʸ = 2x show in exponential form.
6) If f(x)= ax²+ bx + c and f(1)= 3, f(2)=7,
f(3)=13, find the value of a, b, c.
7) If f(x)= a/x + b + cx and f(1)=5, f(-2)=2, f(-1)=-3. Then find the value of f(-3).
EXERCISE - H
E) Show that the following are even
1) 5ˣ + 5⁻ˣ
2) x(eˣ + 1)/(eˣ-1)
3) x² ᶜᵒˢˣ log(x² +1).
4) 1/x log √{x + √(x²+1)}.
4) If f(x)= x² + Kx +1, for all x and if it is even function, find K. 0
EXERCISE - I
F) Show that following are odd
1) x + x³
2) 5ˣ - 5⁻ˣ
3) (eˣ + 1)/(eˣ -1)
4) log {(1+x)/(1-x).
5) log {√(1+x²)+ x}
6) log {√(1+x²) - x}
7) If f(x)= x³ -(k-2)x²+ 2x, for all x and if it is an odd function, find k. 2
EXERCISE - J
G) Find the domain of the following
1) 1/(x²- 3x +2).
2) x²/(x²- 5x +6).
3) (x+3)/(x²- x -2).
4) (x²- 5x +6)/(x²- 8x + 12).
5) 1/(x²- 4).
6) x²/(x²- 25).
7) 1/(x²+1).
8) √(x -10)
9) √(x -a)
10) √(6 -x)
11) √(k -x)
12) √(x² - 5x +6)
13) √(x²- 3x +2)
14) √(x²- 7x +12)
15) √(x²- x - 2)
16) √(x²- 8x +12)
17) √(x²- 4)
18) √(x²- 25)
19) 1/√(x²- 5x +6)
20) 1/√(x²- 7x +12)
21) 1/√(x²- 3x +2)
22) 1/√(x²- 9)
23) 1/√(21-x)
24) 1/√(9- x²)
25) 1/√(16-x²)
26) logₑx
26) logₑ(x-5).
27) logₑ(10- x).
28) logₑ{(4+x)/(4- x)).
29) logₑ(x²- 5x+6)
30) logₑ(x² - 8x+12).
31) logₑ(x² - x-2).
32) logₑ(x² - 10x+21).
EXERCISE- K
Find the range of the following:
1) x/(1+ x²).
2) x²/(1+ x²).
3) x/(x² - 5x +9).
4) (3x-5)/(x² -1).
5) √(4- x²).
MULTIPLE CHOICE QUESTIONS:
1) Let A{1,2,3}, B{2,3,4}, then which of the following is a function from A to B.
A) {(1,2),(1,3),(2,3),(3,3)}
B) {(1,3),(2,4)}. C) {(1,3),(2,2)}
D) {(1,2),(2,3),(3,2),(3,4)}
2) I f f: Q --> Q is defined as f(x)= x², then f⁻¹(9) is equal to
A) 3 B) -3 C) {-3,3} D) ¢
3) which one of the following is not a function ?
A) {(x,y): x,y ∈ R, x² = y}
B) {(x,y): x,y ∈ R, y² = x}
C) {(x,y): x,y ∈ R, x = y³}
D) {(x,y): x,y ∈ R, x³ = y}
4) if f(x)= cos(log x), then
A) f(x²) f(y²) - 1/2 {f(x²/y²) + f(x²y²)} has the value of
A) - 2 B) -1 C) 1/2 D) none
5) If f(x)= cos(log x), then f(x) - {f(x/y)+ f(xy) has the value
A) -1 B) 1/2 C) -2 D) none
6) Let f(x)=| x -1|. Then,
A) f(x²)= [f(x)]² B) f(x+y) =f(x)f(y)
C) f(|x|)= |f(x)| D) none
7) The range of f(x)= cos [x], for -π/2< x< π/3 is
A) {-1,1,0} B) {cos 1, cos 2, 1}
C) {cos 1, - cos 1, 1} D) {-1,1}
8) Which of the following are functions ?
A) {(x,y): y²= x, x,y ∈ R}
B) {(x,y): y= |x|, x,y ∈ R}
C) {(x,y): x²+ y²= 1, x,y ∈ R}
D) {(x,y): x² - y²= 1, x,y ∈ R}
9) If f(x)= log {(1+x)/(1-x)} and g(x)= {(3x+ x³)/(1+ 3x²), then f(g(x)) is equal to
A) f(3x) B) {f(x)}³ C) 3f(x) D) -f(x)
10) If A={1,2,3}, B={x,y}, then the number of functions that can be defined from A into B is
A) 12 B) 8 C) 6 D) 3
11) If f(x)=log {(1+x)/(1-x)}, then f(2x/(1+ x²)) is equal to
A) {f(x)}² B) {f(x)}³ C) 2f(x) D) 3f(x)
12) If f(x)= cos(log x), then value of
A) f(x) f(4)- 1/2 {f(x/4) + f(4x)} is
A) 1 B) -1 C) 0 D) ±1
13) If f(x)= (2ˣ + 2⁻ˣ)/2, then f(x+y) f(x - y) is equals to
A) 1/2 {f(2x)+ f(2y)}
B) 1/2 {f(2x)- f(2y)}
C) 1/4 {f(2x)+ f(2y)}
D) 1/4 {f(2x)- f(2y)}
14) If 2f(x)- 3 f(1/x) = x² (x≠ 0), then f(2) is equal to
A) -7/4 B) 5/2 C) -1 D) none
15) let f: R --> R be defined by f(x)= 2x + |x|. Then f(2x)+ f(-x)- f(x)=
A) 2x B) 2|x| C) -2x D) -2|x|
16) If f(x)= log{(1+x)/(1-x)}, then f(2x/(1+ x²)) is equals to
A) [f(x)]² B) [f(x)]³ C)2f(x) D)3f(x)
17) If x≠ 1 and f(x)= (x+1)/(x-1) is a real function, then f(f(f(2))) is
A) 1 B) 2 C) 3 D) 4
18) If f(x)= cos (log x), then f(1/x) f(1/y) - 1/2 {f(xy) + f(x/y)} is equal to
A) cos(x -y) B) log(cos(x-y))
C) 1 D) cos(x +y)
19) Let f(x)= x, g(x)=1/x and h(x)= f(x)g(x). then, h(x)=1
A) x ∈ R B)x ∈ Q C) x ∈ R - Q
D) x ∈ R, x ≠ 0
20) If f(x)= (sin⁴x + cos²x)/(sin²x + cos⁴x) for x ∈ R, Then f(2002)=
A) 1 B) 2 C) 3 D) 4
21) The function f: R --> R is defined by f(x)= cos²x + sin⁴x. Then, f(R)=
A) {3/4, 1) B) (3/4, 1]
C) [3/4, 1] D) (3/4, 1)
22) Let A={x ∈R: x≠ 0, -4≤ x ≤ 4} and f: A ∈ R be define by f(x) =|x|/x for x∈ A. Then
A) {1, -1} B) {x: 0≤ x ≤ 4}
C) {1} D) { x : -4≤ x ≤ 0}
23) If f: R --> R and g: R --> R are defined by f(x)= 2x+ 3 and g(x)= x² +7, then the values of x such that g(f(x))= 8 are
A) 1,2 B) -1,2 C) -1,-2 D) 1,-2
24) If f: [-2, 2]--> R is Defined by
f(x)= -1, for -2≤ x≤ 0
x- 1, for 0 ≤ x≤ 2 then
{x ∈ [-2,2]: x ≤ 0 and f(|x|)=
A) {-1} B) {0} C) {-1/3} D) ¢
25) ₑf(x)= (10+x)/(20-x), x∈ (-10, 10) and ∈ k {f(200x)/(100+x²))}, then k=
A) 0.5 B) 0.6 C) 0.7 D) 0.8
26) If f is a real valued function given by f(x)= 27x³ + 1/x³ and m, n are the roots of 3x + 1/x = 12. Then,
A) f(m)= f(n) B) f(m)= 10
C) f(n) = -10 D) none
27) If f(x)= 64x³ + 1/x³ and m, n are the roots of 4x + 1/x = 3, then,
A) f(m) = f(n)= -9
B) f(m)= f(n)= 63
C) f(m)≠ f(n) D) none
28) If 3 f(x) + f(1/x) = 1/x - 3 for all non zero x, then f(x)=
A) 1/14(3/x +5x -6)
B) 1/14(- 3/x +5x -6)
C) 1/14(-3/x +5x +6) D) none
29) If f: R--> R be given by f(x) = 4ˣ/(4ˣ+2) for all x belongs to R, then,
A) f(x)= f(1-x) B) f(x)+ f(1-x) = 0
C) f(x)+ f(1-x)= 0 D) f(x)+ f(1-x)=0
30) If f(x)= sin[π²]x + sin[-π²]x, where [x] denotes the greatest integer less than or equal to x, then
A) f(x/2)= 1 B) f(π)= 2
C) f(π)4)= -1 D) none
31) The domain of the function f(x)= √(2- 2x -x²) is
A) [-√3, √3],. B) [-1-√3, -1+ √3]
C) [-2, 2] D) [-2-√3, -2+ √3]
32) the domain of the definition of f(x)= √[(x+3)/{(2-x)(x-5)}] is
A) (-∞,-3) U(2,5)
B) (-∞,3) U(2,5)
C) (-∞,-3] U(][2,5] D) none
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