Problem 91 - College Algebra

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. No quadratic functions have a range of \((-\infty, \infty)\)

Step-by-Step Solution

Answer

The statement 'No quadratic functions have a range of \((-\infty, \infty)\)' is true.

Step 1: Understanding the Quadratic Function's Behavior

Given a quadratic function in the form \(f(x) = ax^2 + bx + c\), if the coefficient \(a\) is positive, then the graph opens upwards. Meaning, it has a minimum point (vertex) and hence the range is \([k, \infty)\) where \(k\) is the y-coordinate of the vertex. If the coefficient \(a\) is negative, then the graph opens downwards. Thus, it has a maximum point (vertex), and the range is \((-\infty, k]\) where \(k\) is the y-coordinate of the vertex.

Step 2: Determining the Statement's Validity

From the previous understanding, it's clear that no quadratic function has a range of \((-\infty, \infty)\) since all quadratic functions have either a maximum or a minimum point hence the range can't extend infinitely in both directions. Consequently, the given statement is true.