13) If [tex]f(x)=2 x-3[/tex] and [tex]g(x)=x^2+1[/tex], then find:
1) [tex](f+g)(x)=2 x-3+x^2+1=[/tex]
2) [tex](f-g)(x)=[/tex]
3) [tex](f g)(x)=[/tex]
4) [tex](\frac{f}{g})(x)=[/tex]
5) [tex](f+g)(-2)=[/tex]
6) [tex](f g)(-2)=[/tex]
7: [tex](f \circ g)(-2)=[/tex]

Find ( f + g ) ( x ) by adding f ( x ) and g ( x ) , resulting in ( f + g ) ( x ) = x 2 + 2 x − 2 .
Find ( f − g ) ( x ) by subtracting g ( x ) from f ( x ) , resulting in ( f − g ) ( x ) = − x 2 + 2 x − 4 .
Find ( f g ) ( x ) by multiplying f ( x ) and g ( x ) , resulting in ( f g ) ( x ) = 2 x 3 − 3 x 2 + 2 x − 3 .
Find ( g f ​ ) ( x ) by dividing f ( x ) by g ( x ) , resulting in ( g f ​ ) ( x ) = x 2 + 1 2 x − 3 ​ .
Evaluate ( f + g ) ( − 2 ) , ( f g ) ( − 2 ) , and ( f ∘ g ) ( − 2 ) , resulting in ( f + g ) ( − 2 ) = − 2 , ( f g ) ( − 2 ) = − 35 , and ( f ∘ g ) ( − 2 ) = 7 .
The final answers are: ( f + g ) ( x ) = x 2 + 2 x − 2 ​ ( f − g ) ( x ) = − x 2 + 2 x − 4 ​ ( f g ) ( x ) = 2 x 3 − 3 x 2 + 2 x − 3 ​ ( g f ​ ) ( x ) = x 2 + 1 2 x − 3 ​ ​ ( f + g ) ( − 2 ) = − 2 ​ ( f g ) ( − 2 ) = − 35 ​ ( f ∘ g ) ( − 2 ) = 7 ​

Explanation

Understanding the Problem We are given two functions, f ( x ) = 2 x − 3 and g ( x ) = x 2 + 1 . Our goal is to find expressions for ( f + g ) ( x ) , ( f − g ) ( x ) , ( f g ) ( x ) , and ( g f ​ ) ( x ) , as well as the values of ( f + g ) ( − 2 ) , ( f g ) ( − 2 ) , and ( f ∘ g ) ( − 2 ) .

Finding (f+g)(x) First, let's find ( f + g ) ( x ) . This is simply the sum of the two functions: ( f + g ) ( x ) = f ( x ) + g ( x ) = ( 2 x − 3 ) + ( x 2 + 1 ) = x 2 + 2 x − 2.

Finding (f-g)(x) Next, let's find ( f − g ) ( x ) . This is the difference between the two functions: ( f − g ) ( x ) = f ( x ) − g ( x ) = ( 2 x − 3 ) − ( x 2 + 1 ) = − x 2 + 2 x − 4.

Finding (fg)(x) Now, let's find ( f g ) ( x ) . This is the product of the two functions: ( f g ) ( x ) = f ( x ) g ( x ) = ( 2 x − 3 ) ( x 2 + 1 ) = 2 x 3 − 3 x 2 + 2 x − 3.

Finding (f/g)(x) Next, let's find ( g f ​ ) ( x ) . This is the quotient of the two functions: ( g f ​ ) ( x ) = g ( x ) f ( x ) ​ = x 2 + 1 2 x − 3 ​ .

Finding (f+g)(-2) Now, let's find ( f + g ) ( − 2 ) . We substitute x = − 2 into the expression for ( f + g ) ( x ) : ( f + g ) ( − 2 ) = ( − 2 ) 2 + 2 ( − 2 ) − 2 = 4 − 4 − 2 = − 2.

Finding (fg)(-2) Next, let's find ( f g ) ( − 2 ) . We substitute x = − 2 into the expression for ( f g ) ( x ) : ( f g ) ( − 2 ) = 2 ( − 2 ) 3 − 3 ( − 2 ) 2 + 2 ( − 2 ) − 3 = 2 ( − 8 ) − 3 ( 4 ) − 4 − 3 = − 16 − 12 − 4 − 3 = − 35.

Finding (f ∘ g)(-2) Finally, let's find ( f ∘ g ) ( − 2 ) . This is f ( g ( − 2 )) . First, we find g ( − 2 ) : g ( − 2 ) = ( − 2 ) 2 + 1 = 4 + 1 = 5. Then, we substitute this into f ( x ) : f ( g ( − 2 )) = f ( 5 ) = 2 ( 5 ) − 3 = 10 − 3 = 7.

Final Answer In summary:



( f + g ) ( x ) = x 2 + 2 x − 2
( f − g ) ( x ) = − x 2 + 2 x − 4
( f g ) ( x ) = 2 x 3 − 3 x 2 + 2 x − 3
( g f ​ ) ( x ) = x 2 + 1 2 x − 3 ​
( f + g ) ( − 2 ) = − 2
( f g ) ( − 2 ) = − 35
( f ∘ g ) ( − 2 ) = 7

Examples
Understanding function operations like addition, subtraction, multiplication, division, and composition is crucial in many real-world applications. For example, in economics, if f ( x ) represents the cost of producing x units and g ( x ) represents the revenue from selling x units, then ( f − g ) ( x ) represents the profit. Similarly, in physics, if f ( t ) represents the position of an object at time t and g ( t ) represents its velocity, then the composition f ( g ( t )) could represent the position of the object as a function of its velocity. These operations allow us to model and analyze complex relationships between different quantities.

Answered by GinnyAnswer | 2025-07-05

We computed various operations involving the functions f ( x ) = 2 x − 3 and g ( x ) = x 2 + 1 . The results include the sum, difference, product, quotient, and evaluations at − 2 . The composition of the functions was also found, providing a comprehensive analysis of their interactions.
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Answered by Anonymous | 2025-07-07