Question

which of the following statements accurately describes a difference between the graphs of ( f(x)=sin(x) ) and ( g(x)=sin(-x) )?
statement #1: if ( (x,y) ) is a point in ( f(x) ), then ( (x,-y) ) is a point in ( g(x) ).
statement #2: the function ( g(x) ) is the same as ( f(x) ).
statement #3: if ( (x,y) ) is a point in ( f(x) ), then ( (-x,-y) ) is a point in ( g(x) ).
(1 point)
statement # is correct.

Explanation:

Step1: Recall the property of sine function

We know that \(\sin(-x)=-\sin(x)\). So \(g(x)=\sin(-x)=-\sin(x)\) and \(f(x)=\sin(x)\).

Step2: Analyze Statement #1

If \((x,y)\) is a point on \(f(x)\) (i.e., \(y = f(x)=\sin(x)\)), then for \(g(x)\), \(g(x)=-\sin(x)\). So when \(x\) is the input for \(g(x)\), \(g(x)=-y\). So the point \((x, - y)\) is on \(g(x)\).

Step3: Analyze Statement #2

Since \(g(x)=-\sin(x)\) and \(f(x)=\sin(x)\), \(g(x)
eq f(x)\) (except when \(\sin(x) = 0\)).

Step4: Analyze Statement #3

If \((x,y)\) is on \(f(x)\) (\(y=\sin(x)\)), then for \(g(x)\), if we substitute \(-x\) into \(g(x)\): \(g(-x)=\sin(-(-x))=\sin(x)=y
eq - y\) (unless \(y = 0\)).

Answer:

Statement #1 is correct.