Question

the table shows three unique functions.

x	f(x)	g(x)	h(x)
-2	4	6	-3
-1	4½	6½	-2½
1	5½	7½	-1½
2	6	8

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

• f(x) has an all negative domain.
• g(x) has the greatest maximum value.
• all three functions share the same range.
• h(x) has a range of all negative numbers.
• all three functions share the same domain.

Explanation:

Step1: Analyze Domain of \( f(x) \)

The domain of a function is the set of \( x \)-values. For \( f(x) \), the \( x \)-values are \(-2, -1, 1, 2\). Since there are positive \( x \)-values (1, 2), the statement " \( f(x) \) has an all negative domain" is false.

Step2: Analyze Maximum Values

Find the maximum values of each function:
- For \( f(x) \): The values are \( 4, 4\frac{1}{2}, 5\frac{1}{2}, 6 \). Maximum is \( 6 \).
- For \( g(x) \): The values are \( 6, 6\frac{1}{2}, 7\frac{1}{2}, 8 \). Maximum is \( 8 \).
- For \( h(x) \): The values are \( -3, -2\frac{1}{2}, -1\frac{1}{2} \). Maximum is \( -1\frac{1}{2} \).
So \( g(x) \) has the greatest maximum value. This statement is true.

Step3: Analyze Ranges

- Range of \( f(x) \): \( \{4, 4\frac{1}{2}, 5\frac{1}{2}, 6\} \) (all positive/non - negative)
- Range of \( g(x) \): \( \{6, 6\frac{1}{2}, 7\frac{1}{2}, 8\} \) (all positive)
- Range of \( h(x) \): \( \{-3, -2\frac{1}{2}, -1\frac{1}{2}\} \) (all negative)
So the ranges are different. The statement "All three functions share the same range" is false.

Step4: Analyze Range of \( h(x) \)

The values of \( h(x) \) are \( -3, -2\frac{1}{2}, -1\frac{1}{2} \), which are all negative. But wait, let's check the domain values for \( h(x) \). The \( x \)-values for \( h(x) \) are \(-2, -1, 1\) (wait, original table: \( x=-2,-1,1 \) for \( h(x) \)? Wait, the table has \( x=-2,-1,1,2 \) for \( f(x) \) and \( g(x) \), but for \( h(x) \), \( x = - 2,-1,1 \) (since the last cell for \( h(x) \) when \( x = 2 \) is empty). Wait, the values of \( h(x) \) are \( -3, -2\frac{1}{2}, -1\frac{1}{2} \), which are all negative. But the statement says " \( h(x) \) has a range of all negative numbers". But is the range only these negative numbers? Yes, but let's check other statements first. Wait, let's go to the domain analysis.

Step5: Analyze Domains

The domain of a function is the set of \( x \)-values. For \( f(x) \), \( g(x) \), and \( h(x) \), the \( x \)-values present are \( -2, -1, 1, 2 \) (for \( f(x) \) and \( g(x) \), and for \( h(x) \), \( x=-2,-1,1 \) (but wait, the table has \( x=-2,-1,1,2 \) columns. Wait, maybe the domain for all three functions, considering the \( x \)-values given (even if \( h(x) \) has no value at \( x = 2 \), but the domain is the set of \( x \)-values for which the function is defined. From the table, the \( x \)-values provided for the functions are \( -2, -1, 1, 2 \) (since the table has rows for \( x=-2,-1,1,2 \)). So for \( f(x) \), \( g(x) \), the domain includes \( -2, -1, 1, 2 \). For \( h(x) \), the domain includes \( -2, -1, 1 \) (since at \( x = 2 \), there's no value). Wait, but maybe the question considers the domain as the set of \( x \)-values in the table (i.e., \( -2, -1, 1, 2 \)) for all three functions (assuming that maybe the empty cell is a typo or we consider the domain as the \( x \)-values given in the table's rows). So:
- Domain of \( f(x) \): \( \{-2, -1, 1, 2\} \)
- Domain of \( g(x) \): \( \{-2, -1, 1, 2\} \)
- Domain of \( h(x) \): If we consider the \( x \)-values in the table, it's \( \{-2, -1, 1, 2\} \) (even if \( h(2) \) is not given, maybe it's part of the domain). So all three functions share the same domain (the set of \( x \)-values in the table: \( -2, -1, 1, 2 \)). This statement is true.

Now, let's re - check the \( h(x) \) range statement. The values of \( h(x) \) are \( -3, -2\frac{1}{2}, -1\frac{1}{2} \), which are all negative, but is the range "all negative numbers"? No, because the range is the set of output values, and these are specific negative numbers, not all real negative numbers. But the other statements: we have two true statements: \( g(x) \) has the greatest maximum value and all three functions share the same domain.

Wait, let's re - evaluate the \( h(x) \) range. The values of \( h(x) \) are \( -3, -2\frac{1}{2}, -1\frac{1}{2} \), which are all negative, but the statement says " \( h(x) \) has a range of all negative numbers". But "all negative numbers" would imply every negative real number, which is not the case. So this statement is false.

So the two true statements are: \( g(x) \) has the greatest maximum value and all three functions share the same domain.

Answer:

B. \( g(x) \) has the greatest maximum value, E. All three functions share the same domain