Math / Algebra

QuestionSuppose that the functions $g$ and $h$ are defined for all real numbers $x$ as follows. $\begin{array}{l} g(x)=4 x^{2} \\ h(x)=x^{3} \end{array}$
Write the expressions for $(g \cdot h)(x)$ and $(g-h)(x)$ and evaluate $(g+h)(-1)$ . $\begin{array}{r} (g \cdot h)(x)=\square \\ (g-h)(x)=\square \\ (g+h)(-1)=\square \end{array}$

Studdy Solution

STEP 1

What is this asking? We're given two functions, $g(x)$ and $h(x)$ , and we need to find the expressions for when they're multiplied and subtracted, and also figure out what happens when we add them and plug in $x = -1$ . Watch out! Don't mix up multiplying functions with multiplying regular numbers!
Also, remember that $(-1)^2$ is positive, but $(-1)^3$ is negative.

STEP 2

1. Find $(g \cdot h)(x)$
2. Find $(g - h)(x)$
3. Evaluate $(g + h)(-1)$

STEP 3

Alright, let's multiply those functions!
We're given $g(x) = 4x^2$ and $h(x) = x^3$ .
So, $(g \cdot h)(x)$ just means $g(x) \cdot h(x)$ .

STEP 4

Let's substitute the expressions for $g(x)$ and $h(x)$ : $(g \cdot h)(x) = (4x^2) \cdot (x^3)$

STEP 5

Now, we multiply the coefficients and add the exponents of $x$ : $(g \cdot h)(x) = 4x^{2+3} = 4x^5$ So, $(g \cdot h)(x) = 4x^5$ !

STEP 6

Time to subtract! $(g - h)(x)$ means $g(x) - h(x)$ .

STEP 7

Let's substitute again: $(g - h)(x) = (4x^2) - (x^3)$

STEP 8

We can rewrite this as: $(g - h)(x) = 4x^2 - x^3$ That's it!
We can't simplify this any further.

STEP 9

Now, we're going to add the functions and then plug in $x = -1$ .

STEP 10

First, let's find the expression for $(g + h)(x)$ : $(g + h)(x) = g(x) + h(x) = 4x^2 + x^3$

STEP 11

Now, let's substitute $x = -1$ : $(g + h)(-1) = 4(-1)^2 + (-1)^3$

STEP 12

Remember, $(-1)^2 = 1$ and $(-1)^3 = -1$ , so: $(g + h)(-1) = 4(1) + (-1) = 4 - 1 = 3$

STEP 13

$(g \cdot h)(x) = 4x^5$ $(g - h)(x) = 4x^2 - x^3$ $(g + h)(-1) = 3$

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Studdy Solution

STEP 1

STEP 2

1. Find $(g \cdot h)(x)$
2. Find $(g - h)(x)$
3. Evaluate $(g + h)(-1)$

STEP 3

Alright, let's multiply those functions!
We're given $g(x) = 4x^2$ and $h(x) = x^3$ .
So, $(g \cdot h)(x)$ just means $g(x) \cdot h(x)$ .

STEP 4

Let's substitute the expressions for $g(x)$ and $h(x)$ : $(g \cdot h)(x) = (4x^2) \cdot (x^3)$

STEP 5

Now, we multiply the coefficients and add the exponents of $x$ : $(g \cdot h)(x) = 4x^{2+3} = 4x^5$ So, $(g \cdot h)(x) = 4x^5$ !

STEP 6

Time to subtract! $(g - h)(x)$ means $g(x) - h(x)$ .

STEP 7

Let's substitute again: $(g - h)(x) = (4x^2) - (x^3)$

STEP 8

We can rewrite this as: $(g - h)(x) = 4x^2 - x^3$ That's it!
We can't simplify this any further.

STEP 9

Now, we're going to add the functions and then plug in $x = -1$ .

STEP 10

First, let's find the expression for $(g + h)(x)$ : $(g + h)(x) = g(x) + h(x) = 4x^2 + x^3$

STEP 11

Now, let's substitute $x = -1$ : $(g + h)(-1) = 4(-1)^2 + (-1)^3$

STEP 12

Remember, $(-1)^2 = 1$ and $(-1)^3 = -1$ , so: $(g + h)(-1) = 4(1) + (-1) = 4 - 1 = 3$

STEP 13

$(g \cdot h)(x) = 4x^5$ $(g - h)(x) = 4x^2 - x^3$ $(g + h)(-1) = 3$

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Studdy Solution

STEP 1

STEP 2

1. Find (g⋅h)(x)(g \cdot h)(x)(g⋅h)(x)2. Find (g−h)(x)(g - h)(x)(g−h)(x)3. Evaluate (g+h)(−1)(g + h)(-1)(g+h)(−1)

STEP 3

Alright, let's **multiply** those functions!We're given g(x)=4x2g(x) = 4x^2g(x)=4x2 and h(x)=x3h(x) = x^3h(x)=x3.So, (g⋅h)(x)(g \cdot h)(x)(g⋅h)(x) just means g(x)⋅h(x)g(x) \cdot h(x)g(x)⋅h(x).

STEP 4

Let's **substitute** the expressions for g(x)g(x)g(x) and h(x)h(x)h(x): (g⋅h)(x)=(4x2)⋅(x3)(g \cdot h)(x) = (4x^2) \cdot (x^3)(g⋅h)(x)=(4x2)⋅(x3)

STEP 5

Now, we **multiply** the coefficients and **add** the exponents of xxx: (g⋅h)(x)=4x2+3=4x5(g \cdot h)(x) = 4x^{2+3} = 4x^5(g⋅h)(x)=4x2+3=4x5 So, (g⋅h)(x)=4x5(g \cdot h)(x) = 4x^5(g⋅h)(x)=4x5!

STEP 6

Time to **subtract**! (g−h)(x)(g - h)(x)(g−h)(x) means g(x)−h(x)g(x) - h(x)g(x)−h(x).

STEP 7

Let's **substitute** again: (g−h)(x)=(4x2)−(x3)(g - h)(x) = (4x^2) - (x^3)(g−h)(x)=(4x2)−(x3)

STEP 8

We can **rewrite** this as: (g−h)(x)=4x2−x3(g - h)(x) = 4x^2 - x^3(g−h)(x)=4x2−x3 That's it!We can't simplify this any further.

STEP 9

Now, we're going to **add** the functions and then **plug in** x=−1x = -1x=−1.

STEP 10

First, let's find the expression for (g+h)(x)(g + h)(x)(g+h)(x): (g+h)(x)=g(x)+h(x)=4x2+x3(g + h)(x) = g(x) + h(x) = 4x^2 + x^3(g+h)(x)=g(x)+h(x)=4x2+x3

STEP 11

Now, let's **substitute** x=−1x = -1x=−1: (g+h)(−1)=4(−1)2+(−1)3(g + h)(-1) = 4(-1)^2 + (-1)^3(g+h)(−1)=4(−1)2+(−1)3

STEP 12

Remember, (−1)2=1(-1)^2 = 1(−1)2=1 and (−1)3=−1(-1)^3 = -1(−1)3=−1, so: (g+h)(−1)=4(1)+(−1)=4−1=3(g + h)(-1) = 4(1) + (-1) = 4 - 1 = 3(g+h)(−1)=4(1)+(−1)=4−1=3

STEP 13

(g⋅h)(x)=4x5(g \cdot h)(x) = 4x^5(g⋅h)(x)=4x5 (g−h)(x)=4x2−x3(g - h)(x) = 4x^2 - x^3(g−h)(x)=4x2−x3 (g+h)(−1)=3(g + h)(-1) = 3(g+h)(−1)=3

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1. Find $(g \cdot h)(x)$
2. Find $(g - h)(x)$
3. Evaluate $(g + h)(-1)$

Alright, let's multiply those functions!
We're given $g(x) = 4x^2$ and $h(x) = x^3$ .
So, $(g \cdot h)(x)$ just means $g(x) \cdot h(x)$ .

Let's substitute the expressions for $g(x)$ and $h(x)$ : $(g \cdot h)(x) = (4x^2) \cdot (x^3)$

Now, we multiply the coefficients and add the exponents of $x$ : $(g \cdot h)(x) = 4x^{2+3} = 4x^5$ So, $(g \cdot h)(x) = 4x^5$ !

Time to subtract! $(g - h)(x)$ means $g(x) - h(x)$ .

Let's substitute again: $(g - h)(x) = (4x^2) - (x^3)$

We can rewrite this as: $(g - h)(x) = 4x^2 - x^3$ That's it!
We can't simplify this any further.

Now, we're going to add the functions and then plug in $x = -1$ .

First, let's find the expression for $(g + h)(x)$ : $(g + h)(x) = g(x) + h(x) = 4x^2 + x^3$

Now, let's substitute $x = -1$ : $(g + h)(-1) = 4(-1)^2 + (-1)^3$

Remember, $(-1)^2 = 1$ and $(-1)^3 = -1$ , so: $(g + h)(-1) = 4(1) + (-1) = 4 - 1 = 3$

$(g \cdot h)(x) = 4x^5$ $(g - h)(x) = 4x^2 - x^3$ $(g + h)(-1) = 3$