# Class 11 RD Sharma Solutions – Chapter 3 Functions – Exercise 3.2

### Question 1. If f(x)=x2-3x+4, then find the values of x satisfying the equation f(x)=f(2x+1).

Solution:

We have,

f(x)=x2-3x+4

Now,

f(2x+1)=(2x+1)2-3(2x+1)+4

f(2x+1)=4x2+1+4x-6x-3+4

f(2x+1)=4x2-2x+2

It is given that

f(x)=f(2x+1)

x2-3x+4=4x2-2x+2

0=4x2-x2-2x+3x+2-4

3x2+x-2=0

3x2+3x-2x-2=0

3x(x+1)-2(x+1)=0

(x+1)(3x-2)=0

x+1=0 or 3x-2=0

x=-1 or x=2/3

Therefore, the value of x are -1 and 2/3.

### Question 2. If f(x)=(x-a)2 (x-b)2 ,find f(a+b).

Solution:

We have,

f(x)=(x-a)2(x-b)2

Now, let us find f(a+b)

f(a+b)=(a+b-a)2(a+b-b)2

f(a+b)=b2a2

Therefore, f(a+b)=(ba)2

### Question 3. If y=f(x)=(ax-b)/(bx-a), show that x=f(y).

Solution:

Given,

y=f(x)=(ax-b)/(bx-a)

As we know, f(y)=(ay-b)/(by-a)

Let us prove that x=f(y).

We have,

y=(ax-b)/(bx-a)

By cross multiplying,

y(bx-a)=ax-b

bxy-ay=ax-b

bxy-ax=ay-b

x(by-a)=ay-b

x=(ay-b)/(by-a)

Therefore, x=f(y)

Hence proved.

### Question 4. If f (x) = 1 / (1 â€“ x), show that f [f {f (x)}] = x.

Solution:

Given:

f (x) = 1 / (1 â€“ x)

Let us prove that f [f {f (x)}] = x.

Firstly, let us solve for f {f (x)}.

f {f (x)} = f {1/(1 â€“ x)}

= 1 / 1 â€“ (1/(1 â€“ x))

= 1 / [(1 â€“ x â€“ 1)/(1 â€“ x)]

= 1 / (-x/(1 â€“ x))

= (1 â€“ x) / -x

= (x â€“ 1) / x

âˆ´ f {f (x)} = (x â€“ 1) / x

Now, we shall solve for f [f {f (x)}]

f [f {f (x)}] = f [(x-1)/x]

= 1 / [1 â€“ (x-1)/x]

= 1 / [(x â€“ (x-1))/x]

= 1 / [(x â€“ x + 1)/x]

= 1 / (1/x)

âˆ´ f [f {f (x)}] = x

Hence proved.

### Question 5. If f (x) = (x + 1) / (x â€“ 1), show that f [f (x)] = x.

Solution:

Given:

f (x) = (x + 1) / (x â€“ 1)

We have to prove that f [f (x)] = x.

f [f (x)] = f [(x+1)/(x-1)]

= [(x+1)/(x-1) + 1] / [(x+1)/(x-1) â€“ 1]

= [[(x+1) + (x-1)]/(x-1)] / [[(x+1) â€“ (x-1)]/(x-1)]

= [(x+1) + (x-1)] / [(x+1) â€“ (x-1)]

= (x+1+x-1)/(x+1-x+1)

= 2x/2

= x

âˆ´ f [f (x)] = x

Hence proved.

### Find:

(i) f (1/2)

(ii) f (-2)

(iii) f (1)

(iv) f (âˆš3)

(v) f (âˆš-3)

Solution:

(i) f (1/2)

When, 0 â‰¤ x â‰¤ 1, f(x) = x

âˆ´ f (1/2) = 1/2

(ii) f (-2)

When, x < 0, f(x) = x2

f (â€“2) = (â€“2)2

= 4

âˆ´ f (â€“2) = 4

(iii) f (1)

When, x â‰¥ 1, f (x) = 1/x

f (1) = 1/1

âˆ´ f(1) = 1

(iv) f (âˆš3)

We have âˆš3 = 1.732 > 1

When, x â‰¥ 1, f (x) = 1/x

âˆ´ f (âˆš3) = 1/âˆš3

(v) f (âˆš-3)

We know âˆš-3 is not a real number and the function f(x) is defined only when x âˆˆ R.

âˆ´ f (âˆš-3) does not exist.

### Question 7. If f(x)=x3-(1/x3), show that f(x)+f(1/x)=0.

Solution:

We have,

f(x)=x3-(1/x)3              —(i)

Now,

f(1/x)=(1/x)3-(1/(1/x)3)

f(1/x)=(1/x)3-x       —(ii)

Adding equation (i) and (ii), we get

f(x)+f(1/x) = (x3-1/x3)+(1/x3-x3

f(x)+f(1/x)=x3-x3+1/x3-1/x3

f(x)+f(1/x)=0

Hence, proved.

### Question 8. If f(x)= 2x/(1+x2),show that f(tan Î¸)=sin 2Î¸.

Solution.

We have,

f(x)=2x/(1+x2)

Now,

f(tan Î¸)=2(tan Î¸)/(1+tan2 Î¸)

f(tan Î¸)=sin 2Î¸    (Because, sin 2Î¸ = 2(tan Î¸)/(1+tan2Î¸))

Hence, proved.

### Question 9. If f(x)=(x-1)/(x+1), then show that

i) f(1/x)=-f(x)

ii) f(1/(-x))=-1/f(x)

Solution.

i) We have,

f(x)=(x-1)/(x+1)

Now,

f(1/x)=((1/x)-1)/((1/x)+1)

f(1/x)=((1-x)/x)/((1+x)/x)

f(1/x)=(1-x)/(1+x)=f(-x)

Hence, proved.

ii) We have,

f(x)=(x-1)/(x+1)

Now,

f(1/(-x))= ((1/(-x))-1)/((1/(-x))+1)

f(1/(-x))=((1+x)/(-x))/((1-x)/(-x))

f(1/(-x))=(1+x)/(1-x)

f(1/(-x))=(-1)/((x-1)/(x+1))

f(1/(-x))=-1/f(x)

Hence, proved.

Solution.

We have,

### Question 11. If for non-zero x, a f(x)+b f(1/x) = 1/x – 5, where a â‰  b, then find f(x).

Solution.

We have,

—(i)

—(ii)

Adding equation (i) and (ii), we get

—(iii)

Subtracting equation (ii) from equation (i)

—(iv)

Adding equations(iii) and (iv), we get

Therefore,

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