StudentAidUnit 4: Functions and Graphs — Exercise 4 1
10th Class Mathematics · Unit 4: Functions and Graphs
4.1.1.If f(x) = 2x + 5, g(x) = 3x - 2, then find: (i) f(x) + g(x) (ii) f(x) - g(x)
Given
Functions
f(x) = 2x+5,quad g(x) = 3x-2
Working
(i) f(x)+g(x)
f(x)+g(x) = (2x+5)+(3x-2)
= 2x+5+3x-2
Result
(i) Result
f(x)+g(x) = 5x+3
Working
(ii) f(x)-g(x)
f(x)-g(x) = (2x+5)-(3x-2)
= 2x+5-3x+2
Result
(ii) Result
f(x)-g(x) = -x+7
4.1.2.If f(x) = x + 2, g(x) = 2x + 4, then find: (i) f(x)g(x) (ii) g(x)f(x) (iii) f(x)/g(x) (iv) g(x)/f(x)
Given
Functions
f(x) = x+2,quad g(x) = 2x+4
Working
(i) f(x)g(x)
f(x)g(x) = (x+2)(2x+4)
= x(2x+4)+2(2x+4) = 2x2+4x+4x+8
Result
(i) Result
f(x)g(x) = 2x2+8x+8
Working
(ii) g(x)f(x)
g(x)f(x) = (2x+4)(x+2)
= 2x(x+2)+4(x+2) = 2x2+4x+4x+8
Result
(ii) Result
g(x)f(x) = 2x2+8x+8
Working
(iii) f(x)/g(x)
f(x)g(x) = x+22x+4
= x+22(x+2)
Result
(iii) Result
f(x)g(x) = 12, x neq -2
Working
(iv) g(x)/f(x)
g(x)f(x) = 2x+4x+2
= 2(x+2)x+2
Result
(iv) Result
g(x)f(x) = 2, x neq -2
4.1.3.For the functions f and g, find: (a) (f∘g)(x) (b) (g∘f)(x) (c) (f∘f)(x) (d) (g∘g)(x)
Given
Functions (as in Q1: f(x)=2x+5, g(x)=3x-2 — the stem does not restate them, they are recovered from the substitutions used)
f(x) = 2x+5,quad g(x) = 3x-2
Formula
(a)
(fcirc g)(x) = f(g(x))
Working
= f(3x-2) = 2(3x-2)+5
= 6x-4+5
Result
(a) Result
(fcirc g)(x) = 6x+1
Formula
(b)
(gcirc f)(x) = g(f(x))
Working
= g(2x+5) = 3(2x+5)-2
= 6x+15-2
Result
(b) Result
(gcirc f)(x) = 6x+13
Formula
(c)
(fcirc f)(x) = f(f(x))
Working
= f(2x+5) = 2(2x+5)+5
= 4x+10+5
Result
(c) Result
(fcirc f)(x) = 4x+15
Formula
(d)
(gcirc g)(x) = g(g(x))
Working
= g(3x-2) = 3(3x-2)-2
= 9x-6-2
Result
(d) Result
(gcirc g)(x) = 9x-8
4.1.4.Find the value of k, such that (f∘g)(x) = (g∘f)(x), where f(x) = 3x + 2, g(x) = 6x - k.
Given
Functions
f(x) = 3x+2,quad g(x) = 6x-k
Formula
(fcirc g)(x) = f(g(x)) = f(6x-k) = 3(6x-k)+2 = 18x-3k+2 quad dots (i)
(gcirc f)(x) = g(f(x)) = g(3x+2) = 6(3x+2)-k = 18x+12-k quad dots (ii)
Given
Condition
(fcirc g)(x) = (gcirc f)(x)
From (i) and (ii)
18x-3k+2 = 18x+12-k
Cancel 18x
-3k+2 = 12-k
Add 3k to both sides
2 = 12+2k
Subtract 12
2-12 = 2k implies -10 = 2k
Result
Result
k = -102 = -5
4.1.5.Given that f(x) = 3x + 2 and g(x) = 2x + 3. Find (i) g(f(4)) (ii) f(f(3)) (iii) f(g(-2))
Given
Functions
f(x) = 3x+2,quad g(x) = 2x+3
Working
(i)
f(4) = 3(4)+2 = 12+2 = 14
g(f(4)) = g(14) = 2(14)+3 = 28+3
Result
(i) Result
g(f(4)) = 31
Working
(ii)
f(3) = 3(3)+2 = 9+2 = 11
f(f(3)) = f(11) = 3(11)+2 = 33+2
Result
(ii) Result
f(f(3)) = 35
Working
(iii)
g(-2) = 2(-2)+3 = -4+3 = -1
f(g(-2)) = f(-1) = 3(-1)+2 = -3+2
Result
(iii) Result
f(g(-2)) = -1
4.1.6.Find f^{-1}(x) in each of the following: (i) f(x) = 2x - 3 (ii) f(x) = 4x^3 - 1 (iii) f(x) = \sqrt{x-1}, x \ge 1 (iv) f(x) = \frac{x+1}{3x-2}, x \neq \frac{2}{3}
Given
(i)
f(x) = 2x-3
Let y=2x-3; interchange x,y: x=2y-3 implies x+3=2y
Result
(i) Result
f-1(x) = x+32
Given
(ii)
f(x) = 4x3-1
Let y=4x3-1; interchange: x=4y3-1 implies y3 = x+14
Result
(ii) Result
f-1(x) = sqrt[3]{x+14
Given
(iii)
f(x) = sqrt{x-1}, x ge 1
Let y=sqrt{x-1} (yge0); square: y2=x-1; interchange: x2=y-1
Result
(iii) Result
f-1(x) = x2+1, x ge 0
Given
(iv)
f(x) = x+13x-2, x neq 23
y=x+13x-2 implies y(3x-2)=x+1 implies 3xy-2y=x+1 implies 3xy-x=1+2y implies x(3y-1)=2y+1 implies x=2y+13y-1
Result
(iv) Result
f-1(x) = 2x+13x-1, x neq 13
4.1.7.The functions f and g are defined such that f(x) = 4x + 2 and g(x) = 6x - 18. (i) Find f^{-1}(x) and g^{-1}(x). (ii) Find x if f^{-1}(x) = g^{-1}(x).
Given
Functions
f(x) = 4x+2,quad g(x) = 6x-18
For f(x)
Let y=4x+2; x = y-24
Result
(i) f^{-1}
f-1(x) = x-24
For g(x)
Let y=6x-18; x = y+186 = y6+3
(i) g^{-1}
g-1(x) = x6+3
Working
(ii)
x-24 = x6+3
Multiply both sides by 12
3(x-2) = 2x+36
3x-6 = 2x+36 implies 3x-2x = 36+6
Result
(ii) Result
x = 42
4.1.8.Verify that f(f^{-1}(x)) = f^{-1}(f(x)) = x for (i) f(x) = x - 6 (ii) f(x) = 7x - 4 (iii) f(x) = \frac{x-3}{4} (iv) f(x) = \frac{x-4}{x+2}, x \neq -2
Given
(i)
f(x) = x-6 implies f-1(x) = x+6
Working
f(f-1(x)) = f(x+6) = (x+6)-6 = x
f-1(f(x)) = f-1(x-6) = (x-6)+6 = x
Result
(i) Result
f(f-1(x)) = f-1(f(x)) = x
Given
(ii)
f(x) = 7x-4 implies f-1(x) = x+47
Working
f(f-1(x)) = 7left(x+47right)-4 = x+4-4 = x
f-1(f(x)) = (7x-4)+47 = 7x7 = x
Result
(ii) Result
f(f-1(x)) = f-1(f(x)) = x
Given
(iii)
f(x) = x-34 implies f-1(x) = 4x+3
Working
f(f-1(x)) = f(4x+3) = (4x+3)-34 = 4x4 = x
f-1(f(x)) = f-1left(x-34right) = 4left(x-34right)+3 = x-3+3 = x
Result
(iii) Result
f(f-1(x)) = f-1(f(x)) = x
Given
(iv)
f(x) = x-4x+2, x neq -2
Find f^{-1}(x)
y=x-4x+2 implies x(y+2)=y-4 implies xy+2x=y-4 implies xy-y=-4-2x implies y(x-1)=-2(x+2)
Result
(iv) f^{-1}
f-1(x) = -2(x+2)x-1, x neq 1
Working
f(f-1(x)) = fleft(-2(x+2)x-1right) = x (simplifies via common denominators)
f-1(f(x)) = f-1left(x-4x+2right) = x (simplifies via common denominators)
Result
(iv) Result
f(f-1(x)) = f-1(f(x)) = x quad Verified.
4.1.9.Without finding f^{-1}(x), find domain and range of f^{-1}(x): (i) f(x) = 12x - 3 (ii) f(x) = \frac{1}{2}x + 8 (iii) f(x) = \frac{x}{1+x}, x \neq -1 (iv) f(x) = \sqrt{x-2}, x \ge 2
Given
Rule
Domain of f-1 = Range of f; quad Range of f-1 = Domain of f
(i)
f(x) = 12x-3 is linear (slope 12 neq 0), so Domain(f)=mathbb{R}, Range(f)=mathbb{R}
Result
(i) Result
Domain(f-1) = mathbb{R},quad Range(f-1) = mathbb{R}
Given
(ii)
f(x) = 12x+8 is linear (slope 12 neq 0), so Domain(f)=mathbb{R}, Range(f)=mathbb{R}
Result
(ii) Result
Domain(f-1) = mathbb{R},quad Range(f-1) = mathbb{R}
Given
(iii)
f(x) = x1+x, x neq -1 implies Domain(f) = mathbb{R}-{-1}, Range(f) = mathbb{R}-{1}
Result
(iii) Result
Domain(f-1) = mathbb{R}-{1},quad Range(f-1) = mathbb{R}-{-1}
Given
(iv)
f(x) = sqrt{x-2}, x ge 2 implies Domain(f) = [2,infty), Range(f) = [0,infty)
Result
(iv) Result
Domain(f-1) = [0,infty),quad Range(f-1) = [2,infty)
4.1.10.Given that f(x) = x^2 + 9 and g(x) = x + 21. Find the values of a such that f(a) = g(a).
Given
Functions
f(x) = x2+9,quad g(x) = x+21
Working
f(a)=g(a) implies a2+9 = a+21
a2-a+9-21 = 0 implies a2-a-12 = 0
Factor
(a-4)(a+3) = 0
Result
Result
a = 4 or a = -3