QuestionCalculate the following:
1. $(g+h)(x)$
2. $(h-f)(x)$
3. $(f-h)(x)$
4. $(g \circ f)(x)$
5. Functions: $f(x)=x-1$ , $g(x)=x^{2}-1$ , $h(x)=3x^{3}-2x-1$

Studdy Solution

STEP 1

Assumptions1. The functions are defined as follows - $f(x)=x-1$ - $g(x)=x^{}-1$ - $h(x)=3 x^{3}- x-1$ . We are required to find the following - $(g+h)(x)$ - $(h-f)(x)$ - $(f-h)(x)$ - $\left(g^{\circ} f\right)(x)$

STEP 2

First, let's find $(g+h)(x)$ . This operation means that we add the functions $g(x)$ and $h(x)$ .
$(g+h)(x) = g(x) + h(x)$

STEP 3

Substitute the given functions $g(x)$ and $h(x)$ into the equation.
$(g+h)(x) = (x^{2}-1) + (3 x^{3}-2 x-1)$

STEP 4

Combine like terms.
$(g+h)(x) =3x^{3} + x^{2} -2x -2$

STEP 5

Next, let's find $(h-f)(x)$ . This operation means that we subtract the function $f(x)$ from $h(x)$ .
$(h-f)(x) = h(x) - f(x)$

STEP 6

Substitute the given functions $h(x)$ and $f(x)$ into the equation.
$(h-f)(x) = (3 x^{3}-2 x-1) - (x-1)$

STEP 7

implify the equation.
$(h-f)(x) =3x^{3} -2x -1 - x +1$

STEP 8

Combine like terms.
$(h-f)(x) =3x^{3} -3x$

STEP 9

Next, let's find $(f-h)(x)$ . This operation means that we subtract the function $h(x)$ from $f(x)$ .
$(f-h)(x) = f(x) - h(x)$

STEP 10

Substitute the given functions $f(x)$ and $h(x)$ into the equation.
$(f-h)(x) = (x-) - (3 x^{3}-2 x-)$

STEP 11

implify the equation.
$(f-h)(x) = x - -3x^{3} +x -$

STEP 12

Combine like terms.
$(f-h)(x) = -x^{} +x -2$

STEP 13

Finally, let's find $\left(g^{\circ} f\right)(x)$ . This operation means that we substitute the function $f(x)$ into $g(x)$ .
$\left(g^{\circ} f\right)(x) = g(f(x))$

STEP 14

Substitute the given function $f(x)$ into $g(x)$ .
$\left(g^{\circ} f\right)(x) = (f(x))^{2}-$

STEP 15

Substitute the given function $f(x)$ into the equation.
$\left(g^{\circ} f\right)(x) = (x-)^{2}-$

STEP 16

Expand the square.
$\left(g^{\circ} f\right)(x) = x^{2} -2x + -$

STEP 17

Combine like terms.
$\left(g^{\circ} f\right)(x) = x^{2} -2x$ So, the solutions are. $(g+h)(x) =3x^{3} + x^{2} -2x -2$
2. $(h-f)(x) =3x^{3} -3x$
3. $(f-h)(x) = -3x^{3} +3x -2$
4. $\left(g^{\circ} f\right)(x) = x^{2} -2x$

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QuestionCalculate the following:
1. $(g+h)(x)$
2. $(h-f)(x)$
3. $(f-h)(x)$
4. $(g \circ f)(x)$
5. Functions: $f(x)=x-1$ , $g(x)=x^{2}-1$ , $h(x)=3x^{3}-2x-1$

Studdy Solution

STEP 1

Assumptions1. The functions are defined as follows - $f(x)=x-1$ - $g(x)=x^{}-1$ - $h(x)=3 x^{3}- x-1$ . We are required to find the following - $(g+h)(x)$ - $(h-f)(x)$ - $(f-h)(x)$ - $\left(g^{\circ} f\right)(x)$

STEP 2

First, let's find $(g+h)(x)$ . This operation means that we add the functions $g(x)$ and $h(x)$ .
$(g+h)(x) = g(x) + h(x)$

STEP 3

Substitute the given functions $g(x)$ and $h(x)$ into the equation.
$(g+h)(x) = (x^{2}-1) + (3 x^{3}-2 x-1)$

STEP 4

Combine like terms.
$(g+h)(x) =3x^{3} + x^{2} -2x -2$

STEP 5

Next, let's find $(h-f)(x)$ . This operation means that we subtract the function $f(x)$ from $h(x)$ .
$(h-f)(x) = h(x) - f(x)$

STEP 6

Substitute the given functions $h(x)$ and $f(x)$ into the equation.
$(h-f)(x) = (3 x^{3}-2 x-1) - (x-1)$

STEP 7

implify the equation.
$(h-f)(x) =3x^{3} -2x -1 - x +1$

STEP 8

Combine like terms.
$(h-f)(x) =3x^{3} -3x$

STEP 9

Next, let's find $(f-h)(x)$ . This operation means that we subtract the function $h(x)$ from $f(x)$ .
$(f-h)(x) = f(x) - h(x)$

STEP 10

Substitute the given functions $f(x)$ and $h(x)$ into the equation.
$(f-h)(x) = (x-) - (3 x^{3}-2 x-)$

STEP 11

implify the equation.
$(f-h)(x) = x - -3x^{3} +x -$

STEP 12

Combine like terms.
$(f-h)(x) = -x^{} +x -2$

STEP 13

Finally, let's find $\left(g^{\circ} f\right)(x)$ . This operation means that we substitute the function $f(x)$ into $g(x)$ .
$\left(g^{\circ} f\right)(x) = g(f(x))$

STEP 14

Substitute the given function $f(x)$ into $g(x)$ .
$\left(g^{\circ} f\right)(x) = (f(x))^{2}-$

STEP 15

Substitute the given function $f(x)$ into the equation.
$\left(g^{\circ} f\right)(x) = (x-)^{2}-$

STEP 16

Expand the square.
$\left(g^{\circ} f\right)(x) = x^{2} -2x + -$

STEP 17

Combine like terms.
$\left(g^{\circ} f\right)(x) = x^{2} -2x$ So, the solutions are. $(g+h)(x) =3x^{3} + x^{2} -2x -2$
2. $(h-f)(x) =3x^{3} -3x$
3. $(f-h)(x) = -3x^{3} +3x -2$
4. $\left(g^{\circ} f\right)(x) = x^{2} -2x$

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QuestionCalculate the following: 1. (g+h)(x)(g+h)(x)(g+h)(x) 2. (h−f)(x)(h-f)(x)(h−f)(x) 3. (f−h)(x)(f-h)(x)(f−h)(x) 4. (g∘f)(x)(g \circ f)(x)(g∘f)(x) 5. Functions: f(x)=x−1f(x)=x-1f(x)=x−1, g(x)=x2−1g(x)=x^{2}-1g(x)=x2−1, h(x)=3x3−2x−1h(x)=3x^{3}-2x-1h(x)=3x3−2x−1

Studdy Solution

STEP 1

STEP 2

First, let's find (g+h)(x)(g+h)(x)(g+h)(x). This operation means that we add the functions g(x)g(x)g(x) and h(x)h(x)h(x).(g+h)(x)=g(x)+h(x)(g+h)(x) = g(x) + h(x)(g+h)(x)=g(x)+h(x)

STEP 3

Substitute the given functions g(x)g(x)g(x) and h(x)h(x)h(x) into the equation.(g+h)(x)=(x2−1)+(3x3−2x−1)(g+h)(x) = (x^{2}-1) + (3 x^{3}-2 x-1)(g+h)(x)=(x2−1)+(3x3−2x−1)

STEP 4

Combine like terms.(g+h)(x)=3x3+x2−2x−2(g+h)(x) =3x^{3} + x^{2} -2x -2(g+h)(x)=3x3+x2−2x−2

STEP 5

Next, let's find (h−f)(x)(h-f)(x)(h−f)(x). This operation means that we subtract the function f(x)f(x)f(x) from h(x)h(x)h(x).(h−f)(x)=h(x)−f(x)(h-f)(x) = h(x) - f(x)(h−f)(x)=h(x)−f(x)

STEP 6

Substitute the given functions h(x)h(x)h(x) and f(x)f(x)f(x) into the equation.(h−f)(x)=(3x3−2x−1)−(x−1)(h-f)(x) = (3 x^{3}-2 x-1) - (x-1)(h−f)(x)=(3x3−2x−1)−(x−1)

STEP 7

implify the equation.(h−f)(x)=3x3−2x−1−x+1(h-f)(x) =3x^{3} -2x -1 - x +1(h−f)(x)=3x3−2x−1−x+1

STEP 8

Combine like terms.(h−f)(x)=3x3−3x(h-f)(x) =3x^{3} -3x(h−f)(x)=3x3−3x

STEP 9

Next, let's find (f−h)(x)(f-h)(x)(f−h)(x). This operation means that we subtract the function h(x)h(x)h(x) from f(x)f(x)f(x).(f−h)(x)=f(x)−h(x)(f-h)(x) = f(x) - h(x)(f−h)(x)=f(x)−h(x)

STEP 10

Substitute the given functions f(x)f(x)f(x) and h(x)h(x)h(x) into the equation.(f−h)(x)=(x−)−(3x3−2x−)(f-h)(x) = (x-) - (3 x^{3}-2 x-)(f−h)(x)=(x−)−(3x3−2x−)

STEP 11

implify the equation.(f−h)(x)=x−−3x3+x−(f-h)(x) = x - -3x^{3} +x -(f−h)(x)=x−−3x3+x−

STEP 12

Combine like terms.(f−h)(x)=−x+x−2(f-h)(x) = -x^{} +x -2(f−h)(x)=−x+x−2

STEP 13

Finally, let's find (g∘f)(x)\left(g^{\circ} f\right)(x)(g∘f)(x). This operation means that we substitute the function f(x)f(x)f(x) into g(x)g(x)g(x).(g∘f)(x)=g(f(x))\left(g^{\circ} f\right)(x) = g(f(x))(g∘f)(x)=g(f(x))

STEP 14

Substitute the given function f(x)f(x)f(x) into g(x)g(x)g(x).(g∘f)(x)=(f(x))2−\left(g^{\circ} f\right)(x) = (f(x))^{2}-(g∘f)(x)=(f(x))2−

STEP 15

Substitute the given function f(x)f(x)f(x) into the equation.(g∘f)(x)=(x−)2−\left(g^{\circ} f\right)(x) = (x-)^{2}-(g∘f)(x)=(x−)2−

STEP 16

Expand the square.(g∘f)(x)=x2−2x+−\left(g^{\circ} f\right)(x) = x^{2} -2x + -(g∘f)(x)=x2−2x+−

STEP 17

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QuestionCalculate the following:
1. $(g+h)(x)$
2. $(h-f)(x)$
3. $(f-h)(x)$
4. $(g \circ f)(x)$
5. Functions: $f(x)=x-1$ , $g(x)=x^{2}-1$ , $h(x)=3x^{3}-2x-1$

First, let's find $(g+h)(x)$ . This operation means that we add the functions $g(x)$ and $h(x)$ .
$(g+h)(x) = g(x) + h(x)$

Substitute the given functions $g(x)$ and $h(x)$ into the equation.
$(g+h)(x) = (x^{2}-1) + (3 x^{3}-2 x-1)$

Combine like terms.
$(g+h)(x) =3x^{3} + x^{2} -2x -2$

Next, let's find $(h-f)(x)$ . This operation means that we subtract the function $f(x)$ from $h(x)$ .
$(h-f)(x) = h(x) - f(x)$

Substitute the given functions $h(x)$ and $f(x)$ into the equation.
$(h-f)(x) = (3 x^{3}-2 x-1) - (x-1)$

implify the equation.
$(h-f)(x) =3x^{3} -2x -1 - x +1$

Combine like terms.
$(h-f)(x) =3x^{3} -3x$

Next, let's find $(f-h)(x)$ . This operation means that we subtract the function $h(x)$ from $f(x)$ .
$(f-h)(x) = f(x) - h(x)$

Substitute the given functions $f(x)$ and $h(x)$ into the equation.
$(f-h)(x) = (x-) - (3 x^{3}-2 x-)$

implify the equation.
$(f-h)(x) = x - -3x^{3} +x -$

Combine like terms.
$(f-h)(x) = -x^{} +x -2$

Finally, let's find $\left(g^{\circ} f\right)(x)$ . This operation means that we substitute the function $f(x)$ into $g(x)$ .
$\left(g^{\circ} f\right)(x) = g(f(x))$

Substitute the given function $f(x)$ into $g(x)$ .
$\left(g^{\circ} f\right)(x) = (f(x))^{2}-$

Substitute the given function $f(x)$ into the equation.
$\left(g^{\circ} f\right)(x) = (x-)^{2}-$

Expand the square.
$\left(g^{\circ} f\right)(x) = x^{2} -2x + -$