Math Snap

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:
$$(g + h)(-1) = 4(1) + (-1) = 4 - 1 = 3$$

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