Question

let (f(x) = 4x + 3) and (g(x) = 4x^2 + 2x). after simplifying,

((fg)(x) = )

question 11

let (f(x) = 3sqrt{x}).

if (g(x)) is the graph of (f(x)) shifted up 1 units and right 3 units, write a formula for (g(x)).

(g(x) = )

enter (sqrt{x}) as sqrt(x).

Explanation:

⚡ Using: Operations with Polynomial Functions · 🆕 New: Function Transformations
Shifting graphs horizontally and vertically

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Part 1: Find \((fg)(x)\)

Step 1: Set up the multiplication

The notation \((fg)(x)\) means we multiply the function \(f(x)\) by the function \(g(x)\):

$$ (fg)(x) = f(x) \cdot g(x) $$

Substitute the given functions:

$$ (fg)(x) = (4x + 3)(4x^2 + 2x) $$

Step 2: Expand and simplify

Distribute each term in the first polynomial to each term in the second polynomial:

$$ (fg)(x) = 4x(4x^2 + 2x) + 3(4x^2 + 2x) $$

$$ (fg)(x) = 16x^3 + 8x^2 + 12x^2 + 6x $$

Combine the like terms (\(8x^2\) and \(12x^2\)):

$$ (fg)(x) = 16x^3 + 20x^2 + 6x $$

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Part 2: Find \(g(x)\)

Step 3: Apply transformations to \(f(x)\)

To shift a function \(f(x)\):

• Right by \(h\) units: Replace \(x\) with \((x - h)\).
• Up by \(k\) units: Add \(k\) to the entire function.

We are given \(f(x) = 3\sqrt{x}\).

• Shifting right 3 units means we replace \(x\) with \((x - 3)\):

$$ 3\sqrt{x - 3} $$

• Shifting up 1 unit means we add \(1\) to the outside:

$$ 3\sqrt{x - 3} + 1 $$

Thus, the formula for \(g(x)\) is:

$$ g(x) = 3\sqrt{x - 3} + 1 $$

Using the requested input format sqrt(x):

$$ g(x) = 3\text{sqrt}(x - 3) + 1 $$

Answer:

• First box: 16x^3 + 20x^2 + 6x
• Second box: 3sqrt(x - 3) + 1