question\nthe graph of the function ( f ) is shown above. if ( a ), ( b ), and ( c ) are values of ( x ) for…

question\nthe graph of the function ( f ) is shown above. if ( a ), ( b ), and ( c ) are values of ( x ) for which ( f(x) ) is defined, which following must be true?\n( \bigcirc f(0)+f(a)=f(a) )\n( \bigcirc ) if ( a < b ), then ( f(a)<f(b) ).\n( \bigcirc ) if ( a < 0 ), then ( f(a)<0 ).\n( \bigcirc ) if ( a + b = c ), then ( f(a)+f(b)=f(c) ).

question\nthe graph of the function ( f ) is shown above. if ( a ), ( b ), and ( c ) are values of ( x ) for which ( f(x) ) is defined, which following must be true?\n( \bigcirc f(0)+f(a)=f(a) )\n( \bigcirc ) if ( a < b ), then ( f(a)<f(b) ).\n( \bigcirc ) if ( a < 0 ), then ( f(a)<0 ).\n( \bigcirc ) if ( a + b = c ), then ( f(a)+f(b)=f(c) ).

Answer

Explanation:

Step1: Analyze (f(0)+f(a)=f(a))

From the graph, when (x = 0), (y=f(0)=0). Then (f(0)+f(a)=0 + f(a)=f(a))

Step2: Analyze "If (a < b), then (f(a)<f(b))"

The function is not strictly increasing for all (x). For example, if (a) is in the negative - (x) region and (b) is in the positive - (x) region, the relationship (f(a)<f(b)) does not hold for all (a < b) (since the function has two different linear - like parts with different slopes)

Step3: Analyze "If (a < 0), then (f(a)<0)"

From the graph, when (x<0), (y = f(x)>0). So the statement "If (a < 0), then (f(a)<0)" is false

Step4: Analyze "If (a + b=c), then (f(a)+f(b)=f(c))"

Let (f(x)=mx) for (x\geq0) and (f(x)=nx) for (x < 0) ((m\neq n)). If (a>0), (b>0), (c=a + b), then (f(a)+f(b)=ma+mb=m(a + b)=f(c)). But if (a<0) and (b>0) (or vice - versa), (f(a)+f(b)=na+mb) and (f(c)=f(a + b)). Since (n\neq m), (na+mb\neq f(a + b)) in general

Answer:

(f(0)+f(a)=f(a))