Function Behaviors and Characteristics
Zero points are where functions cross the x-axis, and different function types have different patterns. Logarithmic functions usually have exactly one zero, while exponential functions of the form f(x) = bˣ typically have no zeros since they never cross the x-axis.
Rational functions display interesting behaviors determined by their numerator and denominator. In a function like f(x) = (x−p)(x−q)(x−r)(x−a)(x−b)(x−c)(x−d), the number of zeros equals the number of distinct factors in the numerator (four in this case), while the number of asymptotes relates to the denominator factors (three vertical asymptotes here).
Piecewise functions behave differently in different regions. For f(x) = {−x,x<0 x,x≥0, we apply the appropriate formula based on the input value. For x = 5, since 5 ≥ 0, we use the second piece, resulting in f(5) = 5.
📈 Visual Insight: Think of rational functions as having "holes" (at asymptotes) and exponential functions as curves that never touch the x-axis but always approach it. These visual patterns can help you sketch graphs quickly!