Question

mathematical modeling
the table shows three unique, discrete functions.
(table with x values: -2, -1, 0, 1, 2; f(x) values: (missing for -2, -1), 1, 4, 10; g(x) values: -4½, -2½, -½, 1½, 3½; h(x) values: (missing for -2), -4, -5, -4, -1)
which statements can be used to accurately compare the functions? choose two correct answers.
options:

• f(x) has the greatest maximum.
• all three functions have an x - intercept.
• the domain of all three functions is the same.
• g(x) has the lowest minimum.

Explanation:

Step1: Analyze \( f(x) \), \( g(x) \), \( h(x) \) values

For \( f(x) \): values are \( 1, 4, 10 \) (positive, increasing). For \( g(x) \): values are \( -4\frac{1}{2}, -2\frac{1}{2}, -\frac{1}{2}, 1\frac{1}{2}, 3\frac{1}{2} \) (increasing). For \( h(x) \): values are \( -4, -5, -4, -1 \) (has a minimum at \( x = 0 \)).

Step2: Check each statement

1. \( f(x) \) has the greatest maximum: \( f(2)=10 \), \( g(2)=3\frac{1}{2} \), \( h(2)=-1 \). So \( f(x) \)’s max is greatest. True.
2. All three have an \( x \)-intercept: \( x \)-intercept is when \( y = 0 \). \( f(x) \) has no \( y = 0 \) (values: \( 1, 4, 10 \) for given \( x \); missing \( x=-2, -1 \) but existing \( x \) have \( f(x)>0 \)). So False.
3. Domain of all three: Domain is \( \{ -2, -1, 0, 1, 2 \} \) for all (since \( x \) values are same for all). True.
4. \( g(x) \) has the lowest minimum: \( g(x) \)’s min is \( -4\frac{1}{2} \), \( h(x) \)’s min is \( -5 \) (lower than \( -4\frac{1}{2} \)). So False.

Answer:

The correct statements are:

• \( f(x) \) has the greatest maximum.
• The domain of all three functions is the same.