Question

the table shows three unique functions.

x	f(x)	g(x)	h(x)
1	0	-6	0
2	13	8	-1
3	26	38	-1.5

options:

• $g(x)$ has the greatest maximum.
• $f(x)$ and $h(x)$ have $x$-intercepts.
• the range of $g(x)$ includes more values than the ranges of $f(x)$ and $h(x)$.
• $h(x)$ has the lowest minimum.

Explanation:

Step1: Analyze \( g(x) \) maximum

Check \( g(x) \) values: \(-10, -6, 8, 38\). Max is \( 38 \). \( f(x) \) max: \( 26 \), \( h(x) \) max: \( 3 \). So \( g(x) \) has greatest max. ✔️

Step2: Check \( x \)-intercepts (\( f(x) \) and \( h(x) \))

\( x \)-intercept when \( y = 0 \). \( f(1) = 0 \), \( h(1) = 0 \). So both have \( x \)-intercepts. ✔️

Step3: Analyze range size

\( f(x) \) values: \(-13, 0, 13, 26\) (range: \(-13\) to \( 26 \)). \( g(x) \) values: \(-10, -6, 8, 38\) (range: \(-10\) to \( 38 \)). \( h(x) \) values: \( 3, 0, -1, -1.5 \) (range: \(-1.5\) to \( 3 \)). \( g(x) \) range is wider? Wait, number of values? No, range is set of \( y \)-values. Wait, the statement says "includes more values" – but range is interval, not count. Wait, maybe misinterpret. Wait, \( g(x) \) values: 4 values, \( f(x) \) 4, \( h(x) \) 4. So this statement is wrong.

Step4: Analyze \( h(x) \) minimum

\( f(x) \) min: \(-13\), \( g(x) \) min: \(-10\), \( h(x) \) min: \(-1.5\). So \( f(x) \) has lowest min, not \( h(x) \). Wrong.

Answer:

• \( g(x) \) has the greatest maximum: Correct (checked values).
• \( f(x) \) and \( h(x) \) have \( x \)-intercepts: Correct (both \( y=0 \) at \( x=1 \)).
• The range of \( g(x) \) includes more values...: Incorrect (same number of values, range interval – \( g(x) \) range is \(-10\) to \( 38 \), \( f(x) \) \(-13\) to \( 26 \), so \( f(x) \) range is wider? Wait, no – \( g(x) \) max is higher, min is higher than \( f(x) \)'s min. Wait, maybe the statement is wrong.
• \( h(x) \) has the lowest minimum: Incorrect (\( f(x) \) min is \(-13\), lower than \( h(x) \)'s \(-1.5\)).

So correct statements:

• \( g(x) \) has the greatest maximum
• \( f(x) \) and \( h(x) \) have \( x \)-intercepts