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 1Find the zero of the linear function f is given by
Solution to Example 1To 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 2Find the zeros of the quadratic function f is given by
Solution to Example 2Solve 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 3Find the zeros of the sine function f is given by
Solution to Example 3Solve 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 = π / 6 + 2 k π and x = 5 π / 6 + 2 k π 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 4Find the zeros of the logarithmic function f is given by
Solution to Example 4Solve 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 5Find the zeros of the exponential function f is given by
Solution to Example 5Solve 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 x1 = square root [ln (3) + 2] and x2 =  square root [ln (3) + 2]
More References and linksApplications, Graphs, Domain and Range of Functions
