Question

1. let ( g ) be an odd function that is strictly increasing. selected values of ( g(x) ) are given in the table above. find the values of the constants ( a, b, ) and ( c ).

( x )	( g(x) )
( -4 )	( -11 )
( -1 )	( a + b )
( 1 )	( c )
( b )	( 11 )
( 12 )	( 17 )
( 17 )	( 23 )

Explanation:

Step1: Recall odd function property

For an odd function \( g(x) \), \( g(-x) = -g(x) \). Also, since \( g \) is strictly increasing, if \( x_1 < x_2 \), then \( g(x_1) < g(x_2) \).

Step2: Find \( a \)

From the table, when \( x = 4 \), \( g(4) = -11 \). Using odd function property, \( g(-4) = -g(4) = 11 \)? Wait, no, wait. Wait, the table has \( x = a \) with \( g(a) = -17 \), and \( x = 12 \) with \( g(12) = 17 \). Since \( g \) is odd, \( g(-12) = -g(12) = -17 \). So \( a = -12 \) because \( g(a) = -17 \) and \( g(-12) = -17 \).

Step3: Find \( b \)

From the table, when \( x = -4 \), \( g(-4) = -11 \)? Wait, no, earlier we saw \( g(4) = -11 \), so \( g(-4) = -g(4) = 11 \)? Wait, no, the table has \( x = b \) with \( g(b) = 11 \). Since \( g(-4) = 11 \) (because \( g(4) = -11 \), so \( g(-4) = -(-11) = 11 \)), so \( b = -4 \)? Wait, no, wait the table: \( x = -4 \) has \( g(-4) = -11 \)? Wait, no, looking at the table again: the row with \( x = 4 \) (wait, no, the table has \( x = -4 \)? Wait, the table rows: \( x \) values are \( a \), \( -4 \), \( -1 \), \( 1 \), \( b \), \( 12 \), \( 17 \). \( g(x) \) values: \( -17 \), \( -11 \), \( a + b \), \( c \), \( 11 \), \( 17 \), \( 23 \). Wait, so \( x = -4 \), \( g(-4) = -11 \); then \( x = 4 \) would be \( g(4) = -g(-4) = 11 \), but in the table, \( x = b \) has \( g(b) = 11 \), so \( b = 4 \)? Wait, no, let's re-express. For odd function, \( g(-x) = -g(x) \). So if \( x = -4 \), \( g(-4) = -11 \), then \( g(4) = -g(-4) = 11 \). So in the table, \( x = b \) has \( g(b) = 11 \), so \( b = 4 \).

Step4: Find \( c \)

For \( x = 1 \) and \( x = -1 \). Since \( g \) is odd, \( g(-1) = -g(1) \). The table has \( x = -1 \), \( g(-1) = a + b \), and \( x = 1 \), \( g(1) = c \). So \( g(-1) = -g(1) \implies a + b = -c \). We found \( a = -12 \), \( b = 4 \), so \( a + b = -12 + 4 = -8 \). Thus, \( -8 = -c \implies c = 8 \). Wait, let's check: \( g(-1) = -8 \), so \( g(1) = -g(-1) = 8 \), so \( c = 8 \).

Wait, let's verify each step:

1. \( a \): \( g(12) = 17 \), so \( g(-12) = -17 \). In the table, \( g(a) = -17 \), so \( a = -12 \). Correct.

1. \( b \): \( g(-4) = -11 \), so \( g(4) = -g(-4) = 11 \). In the table, \( g(b) = 11 \), so \( b = 4 \). Correct.

1. \( c \): \( a = -12 \), \( b = 4 \), so \( a + b = -12 + 4 = -8 \). For \( x = -1 \), \( g(-1) = a + b = -8 \). Since \( g \) is odd, \( g(1) = -g(-1) = -(-8) = 8 \). So \( c = 8 \).

Answer:

Assuming we need to find \( a \), \( b \), \( c \):

\( a = -12 \), \( b = 4 \), \( c = 8 \)

(If the question was to find specific values, but since the original question was cut off, but based on the table, these are the values.)