Drag the tiles to the boxes to form correct pairs. Match each operation involving [tex]f(x)[/tex] and [tex]g(x)[/tex] to its answer.
[tex]f(x)=1-x^2 \text { and } g(x)=\sqrt{11-4 x}[/tex]
[tex]\left(\frac{f}{g}\right)(-1) \quad(g \times f)(2) \quad(g-f)(-1) \quad(g+f)(2)[/tex]
Answer (1)
Evaluate f ( x ) and g ( x ) at x = − 1 and x = 2 .
Calculate ( g f ) ( − 1 ) = g ( − 1 ) f ( − 1 ) = 15 0 = 0 .
Compute ( g × f ) ( 2 ) = g ( 2 ) × f ( 2 ) = 3 × ( − 3 ) = − 3 3 .
Determine ( g − f ) ( − 1 ) = g ( − 1 ) − f ( − 1 ) = 15 − 0 = 15 and ( g + f ) ( 2 ) = g ( 2 ) + f ( 2 ) = 3 − 3 .
The final answers are: ( g f ) ( − 1 ) = 0 , ( g × f ) ( 2 ) = − 3 3 , ( g − f ) ( − 1 ) = 15 , and ( g + f ) ( 2 ) = 3 − 3 .
Explanation
Evaluate the functions First, we need to evaluate the functions f ( x ) and g ( x ) at the given points. We have f ( x ) = 1 − x 2 and g ( x ) = 11 − 4 x .
Calculations at x = -1 Let's start with x = − 1 :
f ( − 1 ) = 1 − ( − 1 ) 2 = 1 − 1 = 0 g ( − 1 ) = 11 − 4 ( − 1 ) = 11 + 4 = 15 ≈ 3.87
Calculations at x = 2 Now, let's evaluate the functions at x = 2 :
f ( 2 ) = 1 − ( 2 ) 2 = 1 − 4 = − 3 g ( 2 ) = 11 − 4 ( 2 ) = 11 − 8 = 3 ≈ 1.73
Evaluate the expressions Now we can evaluate the expressions: ( g f ) ( − 1 ) = g ( − 1 ) f ( − 1 ) = 15 0 = 0 ( g × f ) ( 2 ) = g ( 2 ) × f ( 2 ) = 3 × ( − 3 ) = − 3 3 ≈ − 5.20 ( g − f ) ( − 1 ) = g ( − 1 ) − f ( − 1 ) = 15 − 0 = 15 ≈ 3.87 ( g + f ) ( 2 ) = g ( 2 ) + f ( 2 ) = 3 + ( − 3 ) = 3 − 3 ≈ − 1.27
Final Answer Therefore, the correct pairs are: ( g f ) ( − 1 ) = 0 ( g × f ) ( 2 ) = − 3 3 ≈ − 5.20 ( g − f ) ( − 1 ) = 15 ≈ 3.87 ( g + f ) ( 2 ) = 3 − 3 ≈ − 1.27
Examples
Understanding function operations is crucial in many real-world applications. For instance, in physics, if f ( x ) represents the distance an object travels over time and g ( x ) represents its speed, then ( f / g ) ( x ) could represent the time taken to travel a certain distance at that speed. Similarly, in economics, if f ( x ) is the cost function and g ( x ) is the revenue function, then ( g − f ) ( x ) represents the profit at a given level of production x . These operations help analyze and optimize various processes.
