Solved on Jan 18, 2024

Find the composition of the functions $g(t) = t - 3$ and $f(t) = t^2 + 5t$ .

STEP 1

Assumptions
1. The function $g(t)$ is defined as $g(t) = t - 3$ .
2. The function $f(t)$ is defined as $f(t) = t^2 + 5t$ .
3. We need to find the product of the two functions, denoted as $(g \cdot f)(t)$ .

STEP 2

To find the product of two functions, we multiply them together. The product $(g \cdot f)(t)$ is found by multiplying $g(t)$ by $f(t)$ .
$(g \cdot f)(t) = g(t) \cdot f(t)$

STEP 3

Substitute the expressions for $g(t)$ and $f(t)$ into the product.
$(g \cdot f)(t) = (t - 3) \cdot (t^2 + 5t)$

STEP 4

Use the distributive property to multiply each term in the first polynomial by each term in the second polynomial.
$(g \cdot f)(t) = t \cdot t^2 + t \cdot 5t - 3 \cdot t^2 - 3 \cdot 5t$

STEP 5

Perform the multiplication for each term.
$(g \cdot f)(t) = t^3 + 5t^2 - 3t^2 - 15t$

STEP 6

Combine like terms.
$(g \cdot f)(t) = t^3 + (5t^2 - 3t^2) - 15t$

STEP 7

Simplify the expression by combining the $t^2$ terms.
$(g \cdot f)(t) = t^3 + 2t^2 - 15t$
The product of the functions $g(t)$ and $f(t)$ is $t^3 + 2t^2 - 15t$ .

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Find the composition of the functions $g(t) = t - 3$ and $f(t) = t^2 + 5t$ .

STEP 1

Assumptions
1. The function $g(t)$ is defined as $g(t) = t - 3$ .
2. The function $f(t)$ is defined as $f(t) = t^2 + 5t$ .
3. We need to find the product of the two functions, denoted as $(g \cdot f)(t)$ .

STEP 2

To find the product of two functions, we multiply them together. The product $(g \cdot f)(t)$ is found by multiplying $g(t)$ by $f(t)$ .
$(g \cdot f)(t) = g(t) \cdot f(t)$

STEP 3

Substitute the expressions for $g(t)$ and $f(t)$ into the product.
$(g \cdot f)(t) = (t - 3) \cdot (t^2 + 5t)$

STEP 4

Use the distributive property to multiply each term in the first polynomial by each term in the second polynomial.
$(g \cdot f)(t) = t \cdot t^2 + t \cdot 5t - 3 \cdot t^2 - 3 \cdot 5t$

STEP 5

Perform the multiplication for each term.
$(g \cdot f)(t) = t^3 + 5t^2 - 3t^2 - 15t$

STEP 6

Combine like terms.
$(g \cdot f)(t) = t^3 + (5t^2 - 3t^2) - 15t$

STEP 7

Simplify the expression by combining the $t^2$ terms.
$(g \cdot f)(t) = t^3 + 2t^2 - 15t$
The product of the functions $g(t)$ and $f(t)$ is $t^3 + 2t^2 - 15t$ .

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Find the composition of the functions g(t)=t−3g(t) = t - 3g(t)=t−3 and f(t)=t2+5tf(t) = t^2 + 5tf(t)=t2+5t.

STEP 1

Assumptions1. The function g(t)g(t)g(t) is defined as g(t)=t−3g(t) = t - 3g(t)=t−3.2. The function f(t)f(t)f(t) is defined as f(t)=t2+5tf(t) = t^2 + 5tf(t)=t2+5t.3. We need to find the product of the two functions, denoted as (g⋅f)(t)(g \cdot f)(t)(g⋅f)(t).

STEP 2

To find the product of two functions, we multiply them together. The product (g⋅f)(t)(g \cdot f)(t)(g⋅f)(t) is found by multiplying g(t)g(t)g(t) by f(t)f(t)f(t).(g⋅f)(t)=g(t)⋅f(t)(g \cdot f)(t) = g(t) \cdot f(t)(g⋅f)(t)=g(t)⋅f(t)

STEP 3

Substitute the expressions for g(t)g(t)g(t) and f(t)f(t)f(t) into the product.(g⋅f)(t)=(t−3)⋅(t2+5t)(g \cdot f)(t) = (t - 3) \cdot (t^2 + 5t)(g⋅f)(t)=(t−3)⋅(t2+5t)

STEP 4

Use the distributive property to multiply each term in the first polynomial by each term in the second polynomial.(g⋅f)(t)=t⋅t2+t⋅5t−3⋅t2−3⋅5t(g \cdot f)(t) = t \cdot t^2 + t \cdot 5t - 3 \cdot t^2 - 3 \cdot 5t(g⋅f)(t)=t⋅t2+t⋅5t−3⋅t2−3⋅5t

STEP 5

Perform the multiplication for each term.(g⋅f)(t)=t3+5t2−3t2−15t(g \cdot f)(t) = t^3 + 5t^2 - 3t^2 - 15t(g⋅f)(t)=t3+5t2−3t2−15t

STEP 6

Combine like terms.(g⋅f)(t)=t3+(5t2−3t2)−15t(g \cdot f)(t) = t^3 + (5t^2 - 3t^2) - 15t(g⋅f)(t)=t3+(5t2−3t2)−15t

STEP 7

Simplify the expression by combining the t2t^2t2 terms.(g⋅f)(t)=t3+2t2−15t(g \cdot f)(t) = t^3 + 2t^2 - 15t(g⋅f)(t)=t3+2t2−15tThe product of the functions g(t)g(t)g(t) and f(t)f(t)f(t) is t3+2t2−15tt^3 + 2t^2 - 15tt3+2t2−15t.

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Find the composition of the functions $g(t) = t - 3$ and $f(t) = t^2 + 5t$ .

Assumptions
1. The function $g(t)$ is defined as $g(t) = t - 3$ .
2. The function $f(t)$ is defined as $f(t) = t^2 + 5t$ .
3. We need to find the product of the two functions, denoted as $(g \cdot f)(t)$ .

To find the product of two functions, we multiply them together. The product $(g \cdot f)(t)$ is found by multiplying $g(t)$ by $f(t)$ .
$(g \cdot f)(t) = g(t) \cdot f(t)$

Substitute the expressions for $g(t)$ and $f(t)$ into the product.
$(g \cdot f)(t) = (t - 3) \cdot (t^2 + 5t)$

Use the distributive property to multiply each term in the first polynomial by each term in the second polynomial.
$(g \cdot f)(t) = t \cdot t^2 + t \cdot 5t - 3 \cdot t^2 - 3 \cdot 5t$

Perform the multiplication for each term.
$(g \cdot f)(t) = t^3 + 5t^2 - 3t^2 - 15t$

Combine like terms.
$(g \cdot f)(t) = t^3 + (5t^2 - 3t^2) - 15t$

Simplify the expression by combining the $t^2$ terms.
$(g \cdot f)(t) = t^3 + 2t^2 - 15t$
The product of the functions $g(t)$ and $f(t)$ is $t^3 + 2t^2 - 15t$ .