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Math  /  Algebra

QuestionSuppose that the functions ff and gg are defined for all real numbers xx as follows. f(x)=3x2g(x)=x3\begin{array}{l} f(x)=3 x^{2} \\ g(x)=x^{3} \end{array}
Write the expressions for (f+g)(x)(f+g)(x) and (f−g)(x)(f-g)(x) and evaluate (f⋅g)(−1)(f \cdot g)(-1). (f+g)(x)=(f−g)(x)=(f⋅g)(−1)=\begin{array}{l} (f+g)(x)= \\ (f-g)(x)= \\ (f \cdot g)(-1)= \end{array} □\square □\square □\square

Studdy Solution

STEP 1

1. The functions f(x) f(x) and g(x) g(x) are defined for all real numbers.
2. We need to find expressions for (f+g)(x) (f+g)(x) and (f−g)(x) (f-g)(x) .
3. We need to evaluate (f⋅g)(−1) (f \cdot g)(-1) .

STEP 2

1. Find the expression for (f+g)(x) (f+g)(x) .
2. Find the expression for (f−g)(x) (f-g)(x) .
3. Evaluate (f⋅g)(−1) (f \cdot g)(-1) .

STEP 3

To find (f+g)(x) (f+g)(x) , add the functions f(x) f(x) and g(x) g(x) :
(f+g)(x)=f(x)+g(x)=3x2+x3 (f+g)(x) = f(x) + g(x) = 3x^2 + x^3

STEP 4

To find (f−g)(x) (f-g)(x) , subtract the function g(x) g(x) from f(x) f(x) :
(f−g)(x)=f(x)−g(x)=3x2−x3 (f-g)(x) = f(x) - g(x) = 3x^2 - x^3

STEP 5

To evaluate (f⋅g)(−1) (f \cdot g)(-1) , first find f(−1) f(-1) and g(−1) g(-1) :
f(−1)=3(−1)2=3×1=3 f(-1) = 3(-1)^2 = 3 \times 1 = 3
g(−1)=(−1)3=−1 g(-1) = (-1)^3 = -1
Now, multiply the results:
(f⋅g)(−1)=f(−1)⋅g(−1)=3×(−1)=−3 (f \cdot g)(-1) = f(-1) \cdot g(-1) = 3 \times (-1) = -3
The expressions and evaluation are:
(f+g)(x)=3x2+x3(f−g)(x)=3x2−x3(f⋅g)(−1)=−3\begin{array}{l} (f+g)(x) = 3x^2 + x^3 \\ (f-g)(x) = 3x^2 - x^3 \\ (f \cdot g)(-1) = -3 \end{array}

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