Question

the table shows three unique, discrete functions.

x	f(x)	g(x)	h(x)
-2		-4\\(\frac{1}{2}\\)
-1		-2\\(\frac{1}{2}\\)	-4
0	1	-\\(\frac{1}{2}\\)	-5
1	4	1\\(\frac{1}{2}\\)	-4
2	10	3\\(\frac{1}{2}\\)	-1

which statements can be used to accurately compare the functions? select two options.

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

Answer:

To solve this, we analyze each statement by examining the table for \( f(x) \), \( g(x) \), and \( h(x) \):

Analyzing Each Statement:

1. \( g(x) \) has the lowest minimum:

• For \( g(x) \), the values are \( -4\frac{1}{2}, -2\frac{1}{2}, -\frac{1}{2}, 1\frac{1}{2}, 3\frac{1}{2} \). The minimum is \( -4\frac{1}{2} \).
• For \( h(x) \), the values are \( -4, -5, -4, -1 \). The minimum is \( -5 \) (which is lower than \( -4\frac{1}{2} \)). Thus, this statement is false.

1. \( f(x) \) has the greatest maximum:

• For \( f(x) \), the values are \( 1, 4, 10 \). The maximum is \( 10 \).
• For \( g(x) \), the maximum is \( 3\frac{1}{2} \). For \( h(x) \), the maximum is \( -1 \). Thus, \( f(x) \) has the greatest maximum. This statement is true.

1. All three functions have a \( y \)-intercept:

• The \( y \)-intercept occurs at \( x = 0 \).
• \( f(0) = 1 \), \( g(0) = -\frac{1}{2} \), \( h(0) = -5 \). All three have a value at \( x = 0 \), so they all have a \( y \)-intercept. This statement is true? Wait, no—wait, the problem says "select two options." Wait, let’s check the next statements.

1. All three functions have an \( x \)-intercept:

• An \( x \)-intercept occurs where \( y = 0 \).
• For \( f(x) \): Values are \( 1, 4, 10 \) (all positive; no \( x \)-intercept).
• For \( g(x) \): Values are \( -4\frac{1}{2}, -2\frac{1}{2}, -\frac{1}{2}, 1\frac{1}{2}, 3\frac{1}{2} \) (crosses from negative to positive, but does \( g(x) = 0 \)? From \( x = -1 \) (\( -2\frac{1}{2} \)) to \( x = 0 \) (\( -\frac{1}{2} \)) to \( x = 1 \) (\( 1\frac{1}{2} \))—so \( g(x) \) has an \( x \)-intercept. But \( f(x) \) does not (all values positive). Thus, this statement is false.

1. The domain of all three functions is the same:

• The domain is the set of \( x \)-values. For all three, \( x = -2, -1, 0, 1, 2 \) (though \( f(x) \) and \( h(x) \) have missing values at \( x = -2 \), but the table includes \( x = -1, 0, 1, 2 \) for all? Wait, the table shows \( x = -2, -1, 0, 1, 2 \) for \( g(x) \); for \( f(x) \), \( x = -1, 0, 1, 2 \) (missing \( x = -2 \)); for \( h(x) \), \( x = -1, 0, 1, 2 \) (missing \( x = -2 \))? Wait, no—the table’s first column is \( x \), and each row is \( x \)-value. Wait, the table has \( x = -2, -1, 0, 1, 2 \). For \( f(x) \), the \( x = -2 \) cell is empty, but the domain of a discrete function is the set of \( x \)-values with defined outputs. Wait, the problem says "three unique, discrete functions" with the table showing \( x \)-values \( -2, -1, 0, 1, 2 \). For \( g(x) \), all \( x \)-values have outputs. For \( f(x) \), \( x = -2 \) is empty (no output), so domain of \( f(x) \) is \( \{-1, 0, 1, 2\} \); domain of \( g(x) \) is \( \{-2, -1, 0, 1, 2\} \); domain of \( h(x) \): \( x = -2 \) is empty? Wait, the table for \( h(x) \) at \( x = -2 \) is empty, so \( h(x) \)’s domain is \( \{-1, 0, 1, 2\} \). Wait, this is confusing. But the intended interpretation is likely that the domain (set of \( x \)-values considered) is \( \{-2, -1, 0, 1, 2\} \) for all, even if some cells are empty (maybe a typo). Assuming that, the domain is the same. But let’s recheck the other statements.

Wait, the problem says "select two options." Let’s re-express:

• Statement 2 (\( f(x) \) has the greatest maximum): True, as \( f(x) \)’s maximum is \( 10 \), higher than \( g(x) \)’s \( 3\frac{1}{2} \) and \( h(x) \)’s \( -1 \).
• Statement 5 (The domain of all three functions is the same): The \( x \)-values in the table are \( -2, -1, 0, 1, 2 \) for all three (even if some cells are empty, the domain of a discrete function is the set of \( x \)-values with defined inputs). For \( f(x) \), \( x = -2 \) is empty, but maybe the table implies \( x \in \{-2, -1, 0, 1, 2\} \) for all. Alternatively, maybe the empty cells are just not filled, but the domain is the same.

Wait, let’s re-verify:

• \( f(x) \) values: \( x = -1 \) (empty? No, \( f(-1) \) is empty? Wait, the table:

• \( x = -2 \): \( f(x) \) empty, \( g(x) = -4\frac{1}{2} \), \( h(x) \) empty.
• \( x = -1 \): \( f(x) \) empty? No, wait the table:

Wait, the original table:

\( x \)	\( f(x) \)	\( g(x) \)	\( h(x) \)
\( -1 \)	(empty)	\( -2\frac{1}{2} \)	\( -4 \)
\( 0 \)	\( 1 \)	\( -\frac{1}{2} \)	\( -5 \)
\( 1 \)	\( 4 \)	\( 1\frac{1}{2} \)	\( -4 \)
\( 2 \)	\( 10 \)	\( 3\frac{1}{2} \)	\( -1 \)

Ah, so \( f(x) \) has values at \( x = 0, 1, 2 \) (and \( x = -1 \) is empty? Wait, \( x = -1 \) for \( f(x) \) is empty? No, maybe the table is misformatted. Wait, the problem says "three unique, discrete functions"—so \( f(x) \) has outputs at \( x = 0, 1, 2 \) (maybe \( x = -1 \) and \( x = -2 \) are not in its domain). \( g(x) \) has outputs at \( x = -2, -1, 0, 1, 2 \). \( h(x) \) has outputs at \( x = -1, 0, 1, 2 \) ( \( x = -2 \) is empty).

But the key is to find two true statements. Let’s re-express:

• \( f(x) \) has the greatest maximum: \( f(x) \) values are \( 1, 4, 10 \) (max 10). \( g(x) \) max is \( 3\frac{1}{2} \), \( h(x) \) max is \( -1 \). So this is true.
• The domain of all three functions is the same: If we consider the \( x \)-values in the table ( \( -2, -1, 0, 1, 2 \) ), even if some have empty cells, maybe the domain is the set of \( x \)-values listed (assuming the empty cells are just not filled, but the functions are defined for those \( x \)-values). Alternatively, maybe the problem considers the domain as \( \{-2, -1, 0, 1, 2\} \) for all, since the table includes those \( x \)-values. Thus, this statement is true.

Final Answer:

The two correct options are:

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

(Note: Depending on interpretation, "All three functions have a \( y \)-intercept" is also true, but the problem says "select two options." The most consistent pair is \( f(x) \) has the greatest maximum and The domain of all three functions is the same.)

So the correct options are:

• \( \boldsymbol{f(x) \text{ has the greatest maximum}} \)
• \( \boldsymbol{\text{The domain of all three functions is the same}} \)