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Math  /  Algebra

Questionand proaluct of two functions
Suppose that the functions gg and hh are defined for all real numbers xx as follows. g(x)=2x2h(x)=4x\begin{array}{l} g(x)=2 x^{2} \\ h(x)=4 x \end{array}
Write the expressions for (g⋅h)(x)(g \cdot h)(x) and (g−h)(x)(g-h)(x) and evaluate (g+h)(−1)(g+h)(-1). (g∗h)(x)=□(g−h)(x)=□(g+h)(−1)=□\begin{aligned} \left(g^{*} h\right)(x) & =\square \\ (g-h)(x) & =\square \\ (g+h)(-1) & =\square \end{aligned} □\square

Studdy Solution

STEP 1

1. We are given two functions g(x)=2x2 g(x) = 2x^2 and h(x)=4x h(x) = 4x .
2. We need to find expressions for the product and difference of these functions.
3. We need to evaluate the sum of these functions at x=−1 x = -1 .

STEP 2

1. Find the expression for the product of the functions (g⋅h)(x) (g \cdot h)(x) .
2. Find the expression for the difference of the functions (g−h)(x) (g - h)(x) .
3. Evaluate the sum of the functions at x=−1 x = -1 , (g+h)(−1) (g + h)(-1) .

STEP 3

Calculate the product of the functions g(x) g(x) and h(x) h(x) :
(g⋅h)(x)=g(x)×h(x)=(2x2)×(4x) (g \cdot h)(x) = g(x) \times h(x) = (2x^2) \times (4x)
=8x3 = 8x^3

STEP 4

Calculate the difference of the functions g(x) g(x) and h(x) h(x) :
(g−h)(x)=g(x)−h(x)=2x2−4x (g - h)(x) = g(x) - h(x) = 2x^2 - 4x

STEP 5

Evaluate the sum of the functions at x=−1 x = -1 :
(g+h)(x)=g(x)+h(x)=2x2+4x (g + h)(x) = g(x) + h(x) = 2x^2 + 4x
Substitute x=−1 x = -1 :
(g+h)(−1)=2(−1)2+4(−1) (g + h)(-1) = 2(-1)^2 + 4(-1)
=2(1)−4 = 2(1) - 4
=2−4 = 2 - 4
=−2 = -2
The expressions and evaluation are:
(g∗h)(x)=8x3(g−h)(x)=2x2−4x(g+h)(−1)=−2\begin{aligned} \left(g^{*} h\right)(x) & = 8x^3 \\ (g-h)(x) & = 2x^2 - 4x \\ (g+h)(-1) & = -2 \end{aligned}

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