How to find the zeros of
functions; tutorial with examples and detailed solutions. The zeros of a function f are found by solving the equation f(x) = 0.

__Example 1:__ Find the zero of the linear function f is given by

**f(x) = -2 x + 4**
__Solution to Example 1__

To find the zeros of function f, solve the equation

f(x) = -2x + 4 = 0

Hence the zero of f is give by

x = 2

__Example 2:__ Find the zeros of the quadratic function f is given by

**f(x) = -2 x**^{ 2} - 5 x + 7
__Solution to Example 2__

Solve f(x) = 0

f(x) = -2 x^{ 2} - 5 x + 7 = 0

Factor the expression -2 x^{ 2} - 6 x + 8

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

and solve for x

x = -7 / 2 and x = 1

The graph of function f is shown below. The zeros of a function are the x coordinates of the x intercepts of the graph of f.

__Example 3:__ Find the zeros of the sine function f is given by

**f(x) = sin(x) - 1 / 2**
__Solution to Example 3__

Solve f(x) = 0

sin (x) - 1 / 2 = 0

Rewrite as follows

sin (x) = 1 / 2

The above equation is a trigonometric equation and has an infinite number of solutions given by

x = Pi / 6 + 2 k Pi and x = 5 Pi / 6 + 2 k Pi where k is any integer taking the values 0 , 1, -1, 2, -2 ...

The graph of f is shown below. The number of zeros of function f defined by f(x) = sin(x) - 1 / 2 are is infinite simply because function f is periodic.

__Example 4:__ Find the zeros of the logarithmic function f is given by

**f(x) = ln (x - 3) - 2**
__Solution to Example 4__

Solve f(x) = 0

ln (x - 3) - 2 = 0

Rewrite as follows

ln (x - 3) = 2

Rewrite the above equation changing it from logarithmic to exponential form

x - 3 = e^{ 2}

and solve to find one zero

x = 3 + e^{ 2}

__Example 5:__ Find the zeros of the exponential function f is given by

**f(x) = e**^{x2 - 2} - 3
__Solution to Example 4__

Solve f(x) = 0

e^{x2 - 2} - 3 = 0

Rewrite the above equation as follows

e^{x2 - 2} = 3

Rewrite the above equation changing it from exponential to logarithmic form

x^{2} - 2 = ln (3)

Solve the above equation to find two zeros of f

x = square root [ln (3) + 2] and x = - square root [ln (3) + 2]

More tutorials on functions.

Applications, Graphs, Domain and Range of Functions