Question

if \\(f(x)\\) is a linear function and the domain of \\(f(x)\\) is the set of all real numbers, which statement cannot be true?

the graph of \\(f(x)\\) has zero \\(x\\)-intercepts.
the graph of \\(f(x)\\) has exactly one \\(x\\)-intercept.
the graph of \\(f(x)\\) has exactly two \\(x\\)-intercepts.
the graph of \\(f(x)\\) has infinitely many \\(x\\)-intercepts.

Explanation:

Define the general form of a linear function

$$ f(x) = mx + b $$

Analyze the possible number of x-intercepts

$$
LATEXBLOCK0
$$

Determine the impossible statement

$$ \text{A linear function } f(x) = mx + b \text{ cannot have exactly two } x\text{-intercepts.} $$

Answer:

• The graph of \(f(x)\) has zero \(x\)-intercepts.
• The graph of \(f(x)\) has exactly one \(x\)-intercept.
• The graph of \(f(x)\) has exactly two \(x\)-intercepts. (Correct answer)
• The graph of \(f(x)\) has infinitely many \(x\)-intercepts.