How To Find Zeros Of A Function - Free Mathematics Tutorials

How to Find Zeros of Functions

The zeros of a function \( f(x) \) are the values of \( x \) for which \( f(x) = 0 \). In other words, they are the x-coordinates of the points where the graph of the function intersects the x-axis.

Example 1: Linear Function

Find the zero of the linear function:

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

Solution

Set \( f(x) = 0 \) and solve for \( x \):

\( -2x + 4 = 0 \)

\( x = 2 \)

Example 2: Quadratic Function

Find the zeros of the quadratic function:

\( f(x) = -2x^2 - 5x + 7 \)

Solution

Set \( f(x) = 0 \) and solve for \( x \):

\( -2x^2 - 5x + 7 = 0 \)

Factor or use the quadratic formula:

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

Thus, the zeros are:

\( x = -\frac{7}{2} \) and \( x = 1 \)

Example 3: Sine Function

Find the zeros of the trigonometric function:

\( f(x) = \sin(x) - \frac{1}{2} \)

Solution

Set \( f(x) = 0 \) and solve:

\( \sin(x) - \frac{1}{2} = 0 \)

\( \sin(x) = \frac{1}{2} \)

The solutions are:

\( x = \frac{\pi}{6} + 2k\pi \) and \( x = \frac{5\pi}{6} + 2k\pi \), where \( k \) is any integer.

Example 4: Logarithmic Function

Find the zero of the logarithmic function:

\( f(x) = \ln(x - 3) - 2 \)

Solution

Set \( f(x) = 0 \) and solve:

\( \ln(x - 3) - 2 = 0 \)

\( \ln(x - 3) = 2 \)

Convert to exponential form:

\( x - 3 = e^2 \)

Thus, the zero is:

\( x = 3 + e^2 \)

Example 5: Exponential Function

Find the zeros of the exponential function:

\( f(x) = e^{x^2 - 2} - 3 \)

Solution

Set \( f(x) = 0 \) and solve:

\( e^{x^2 - 2} - 3 = 0 \)

\( e^{x^2 - 2} = 3 \)

Take the natural logarithm:

\( x^2 - 2 = \ln(3) \)

Thus, the zeros are:

\( x = \sqrt{\ln(3) + 2} \) and \( x = -\sqrt{\ln(3) + 2} \)

Additional References

Applications, Graphs, Domain and Range of Functions