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
View Step-by-Step Solution
Set \( f(x) = 0 \) and solve for \( x \):
\[ -2x + 4 = 0 \] \[ 2x = 4 \implies x = 2 \]Thus, the zero is \( x = 2 \).
Example 2: Quadratic Function
Find the zeros of the quadratic function:
\[ f(x) = -2x^2 - 5x + 7 \]Solution
View Step-by-Step Solution
Set \( f(x) = 0 \) and solve for \( x \):
\[ -2x^2 - 5x + 7 = 0 \]Factoring the quadratic expression:
\[ (-2x - 7)(x - 1) = 0 \]Thus, the zeros are:
\[ x = -\frac{7}{2} \quad \text{and} \quad x = 1 \]
Example 3: Sine Function
Find the zeros of the trigonometric function:
\[ f(x) = \sin(x) - \frac{1}{2} \]Solution
View Step-by-Step Solution
Set \( f(x) = 0 \) and solve:
\[ \sin(x) - \frac{1}{2} = 0 \implies \sin(x) = \frac{1}{2} \]The general solutions over the domain are:
\[ x = \frac{\pi}{6} + 2k\pi \quad \text{and} \quad x = \frac{5\pi}{6} + 2k\pi \]where \( k \) is any integer.
Example 4: Logarithmic Function
Find the zero of the logarithmic function:
\[ f(x) = \ln(x - 3) - 2 \]Solution
View Step-by-Step Solution
Set \( f(x) = 0 \) and solve:
\[ \ln(x - 3) - 2 = 0 \implies \ln(x - 3) = 2 \]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
View Step-by-Step Solution
Set \( f(x) = 0 \) and solve:
\[ e^{x^2 - 2} - 3 = 0 \implies e^{x^2 - 2} = 3 \]Take the natural logarithm of both sides:
\[ x^2 - 2 = \ln(3) \implies x^2 = \ln(3) + 2 \]Thus, the zeros are:
\[ x = \sqrt{\ln(3) + 2} \quad \text{and} \quad x = -\sqrt{\ln(3) + 2} \]