Practice Questions on Methods to Find Zeros of Functions

Q1. Find zeroes of the following function f(x) = 7x² – 9x + 2.

Solution:

Given Function f(x) = 7x² – 9x + 2,

Factorize the given function

f(x) = 7x² – 7x -2x + 2

f(x) = 7x(x-1) -2 (x-1)

f(x) = (x-1)(7x-2)

For Finding zeros put f(x) =0,

or (x-1)(7x-2) =0

or (x-1) =0 or ( 7x -2 ) = 0

or x=1 or x = 2/7

So Zeroes of the function are {1,2/7}.

Q2. Find zeroes of the function f(x) = tan²x -2tanx – 3, in the interval [0, π/2].

Solution:

Given f(x) = tan²x – 2tanx – 3,

or let tanx = p

So f(x) becomes

= p² -2p – 3

Which is Quadratic Equation in one variable and can be solved using the quadratic formula

p = (2 +√2² + 4×3 ) / 2 or p = (2 -√2² + 4×3 ) / 2

p = ( 2 + √16) /2 or p = (2 -√16 ) / 2

p = 6/2 or p = -2/2

p = 3 or -1

So tanx = 3 or tanx =-1

Hence tan^-1(3) is the Zero of function in the interval [0, π/2] as tan^-1(-1) lies out of the interval [0, π/2].

Q3. Find the roots of the function f(x) = x² – 4 in the interval x ∈ [ 0,8] Using Bisection Method.

Solution:

Given f(x) = x² – 4,

Let a = 0 and b = 0

1st iteration:

Mid point of and b is m = a+b /2 = 0+8 / 2 = 4

and f(m) = (4)² – 4 = 12 ≠ 0

So 4 is not a Zero of the function

2nd Iteration:

Since f(m) > 0 so we take the interval [0, 4]

Where

a = 0, b =4 and mid point of a and b is = (0 + 4) / 2 = 2

And

f(m) = (2)² -4 = 0

So 2 is zero of the given function f(x) = x² -4

Q4. Find the zeroes of the function f(x) = x³ -5x² + 11x – 6.

Solution:

Given f(x) = x³ -5x² + 11x – 6

f(x) = (x-1) (x² -5x + 6 )

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

put f(x)=0

either x=1, 2, 3 for f(x)=0

So x =1, 2, 3 are the required zeros of the given function

Zeroes of a function are those real, complex or imaginary values when put in the function the value of the function becomes zero. There are multiple methods to find zeroes of function according to the function type. Zeroes of a function are very important while describing that function or while plotting the graph of that function.

In this article, we will see different Methods to find zeros of functions.