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