Which statements are true for the functions [tex]g(x)=x^2[/tex] and [tex]h(x)=-x^2[/tex]? Check all that apply.
A. For any value of [tex]x[/tex], [tex]g(x)[/tex] will always be greater than [tex]h(x)[/tex].
B. For any value of [tex]x[/tex], [tex]h(x)[/tex] will always be greater than [tex]g(x)[/tex].
C. [tex]g(x)\ \textgreater \ h(x)[/tex] for [tex]x=-1[/tex].
D. [tex]g(x)\ \textless \ h(x)[/tex] for [tex]x=3[/tex].
E. For positive values of [tex]x[/tex], [tex]g(x)\ \textgreater \ h(x)[/tex].
F. For negative values of [tex]x[/tex], [tex]g(x)\ \textgreater \ h(x)[/tex].
Answer (2)
Statement 3 is true because at x = − 1 , g ( − 1 ) = 1 and h ( − 1 ) = − 1 , so h(-1)"> g ( − 1 ) > h ( − 1 ) .
Statement 5 is true because for positive x , x 2 is positive and − x 2 is negative, thus h(x)"> g ( x ) > h ( x ) .
Statement 6 is true because for negative x , x 2 is positive and − x 2 is negative, thus h(x)"> g ( x ) > h ( x ) .
The true statements are: Statement 3, Statement 5, and Statement 6, so the answer is h(x) \text{ for } x=-1; \text{ For positive values of } x, g(x)>h(x); \text{ For negative values of } x, g(x)>h(x)}"> g ( x ) > h ( x ) for x = − 1 ; For positive values of x , g ( x ) > h ( x ) ; For negative values of x , g ( x ) > h ( x ) .
Explanation
Problem Analysis We are given two functions, g ( x ) = x 2 and h ( x ) = − x 2 , and we need to determine which of the given statements are true. Let's analyze each statement one by one.
Analyzing Statement 1 Statement 1: For any value of x , g ( x ) will always be greater than h ( x ) . This means -x^2"> x 2 > − x 2 for all x . Let's test this with x = 0 . If x = 0 , then g ( 0 ) = 0 2 = 0 and h ( 0 ) = − 0 2 = 0 . Since 0 is not greater than 0 , this statement is false.
Analyzing Statement 2 Statement 2: For any value of x , h ( x ) will always be greater than g ( x ) . This means x^2"> − x 2 > x 2 for all x . Let's test this with x = 1 . If x = 1 , then g ( 1 ) = 1 2 = 1 and h ( 1 ) = − 1 2 = − 1 . Since − 1 is not greater than 1 , this statement is false.
Analyzing Statement 3 Statement 3: h(x)"> g ( x ) > h ( x ) for x = − 1 . This means h(-1)"> g ( − 1 ) > h ( − 1 ) . We have g ( − 1 ) = ( − 1 ) 2 = 1 and h ( − 1 ) = − ( − 1 ) 2 = − 1 . Since -1"> 1 > − 1 , this statement is true.
Analyzing Statement 4 Statement 4: g ( x ) < h ( x ) for x = 3 . This means g ( 3 ) < h ( 3 ) . We have g ( 3 ) = ( 3 ) 2 = 9 and h ( 3 ) = − ( 3 ) 2 = − 9 . Since 9 is not less than − 9 , this statement is false.
Analyzing Statement 5 Statement 5: For positive values of h(x)"> x , g ( x ) > h ( x ) . This means -x^2"> x 2 > − x 2 for all 0"> x > 0 . Since x is positive, x 2 is positive and − x 2 is negative. Any positive number is greater than any negative number, so this statement is true.
Analyzing Statement 6 Statement 6: For negative values of h(x)"> x , g ( x ) > h ( x ) . This means -x^2"> x 2 > − x 2 for all x < 0 . Since x is negative, x 2 is positive and − x 2 is negative. Any positive number is greater than any negative number, so this statement is true.
Final Answer Therefore, the true statements are: h(x)"> g ( x ) > h ( x ) for x = − 1 , For positive values of h(x)"> x , g ( x ) > h ( x ) , and For negative values of h(x)"> x , g ( x ) > h ( x ) .
Examples
Understanding functions like g ( x ) = x 2 and h ( x ) = − x 2 helps us model various real-world phenomena. For instance, the power output of a solar panel might be modeled by a function similar to g ( x ) , where x represents the intensity of sunlight. Conversely, h ( x ) could represent the energy consumption of a device that increases with usage, but negatively impacts battery life. Analyzing these functions allows us to optimize solar panel efficiency or manage battery usage effectively. The relationship between these functions can also be visualized on a graph, providing a clear picture of their behavior.
The true statements regarding the functions g ( x ) = x 2 and h ( x ) = − x 2 are: C ( h(-1)"> g ( − 1 ) > h ( − 1 ) ), E (for positive x , h(x)"> g ( x ) > h ( x ) ), and F (for negative x , h(x)"> g ( x ) > h ( x ) ).
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