Problem 91 - Algebra and Trigonometry

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: Analyzing the statement

The given statement is 'No quadratic functions have a range of \((-\infty, \infty)\)'. This implies that all quadratic functions do not have a range that covers all real numbers from negative infinity to positive infinity.

Step 2: Understanding quadratic functions

A quadratic function in the form of \(f(x) = ax^2 + bx + c\) is a parabola. If \(a > 0\), the parabola opens upward and the range is \([k, \infty)\), where \(k\) is the y-coordinate of the vertex. If \(a < 0\), the parabola opens downward and the range is \((-\infty, k]\), where \(k\) is the y-coordinate of the vertex.

Step 3: Verifying the statement

Since all quadratic functions are of the form of a parabola and parabolas do not have a range of \((-\infty, \infty)\), the statement 'No quadratic functions have a range of \((-\infty, \infty)\)' is true.