Question

how many roots of \\(f(x)\\) are rational numbers?

1
2
4
6

Explanation:

Identify the x-intercepts from the graph

The roots of the function \(f(x)\) correspond to the \(x\)-intercepts of the graph where \(f(x) = 0\). Looking at the graph, the curve crosses the \(x\)-axis at four distinct points:

• One point is between \(x = -2\) and \(x = -1\), specifically at \(x = -1.5\) or \(-\frac{3}{2}\).
• One point is between \(x = -1\) and \(x = 0\), specifically at \(x = -0.5\) or \(-\frac{1}{2}\).
• One point is between \(x = 0\) and \(x = 1\), specifically at \(x = 0.5\) or \(\frac{1}{2}\).
• One point is between \(x = 1\) and \(x = 2\), specifically at \(x = 1.5\) or \(\frac{3}{2}\).

Determine which roots are rational numbers

A rational number is any number that can be expressed as the quotient or fraction \(\frac{p}{q}\) of two integers.
The identified \(x\)-intercepts are:

$$ x = -1.5 = -\frac{3}{2} $$

$$ x = -0.5 = -\frac{1}{2} $$

$$ x = 0.5 = \frac{1}{2} $$

$$ x = 1.5 = \frac{3}{2} $$

All four values are exact terminating decimals that can be written as fractions of integers.

Count the rational roots

Since all 4 real roots are rational numbers, the total number of rational roots is 4.

Answer:

• 1
• 2
• 4 (Correct answer)
• 6