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What is the result when \( (3x^3 + x) – (2x^3 + 4x) \) is simplified?

For the function \( f(x) = \frac{1}{x} \), replacing it with \( \frac{1}{x + 3} \) results in:

Simplify the following rational expression: \[\frac{x^3 + 3x^2 + 2x}{x^2 + x}\]

What are the intervals where the function is less than zero? The x-axis spans from below negative 4 to 4, and the y-axis spans from below 0 to above 40. The x-axis has a scale of 2 in increments of 0.5 and the y-axis has a scale of 20 in increments of 5. The purple curve represents a polynomial function with three turning points. It starts from the bottom left of the third quadrant, rises to a local maximum near (negative 3, 45), decreases to a local minimum around the origin (0, 0). It rises again to another local maximum at a coordinate with x value roughly halfway between 1 and 1.5 and y value roughly between 5 and 10 and then decreases sharply as x increases.

Which transformation is represented by \( g(x) = \vert x \vert – 7 \)?

How does the graph of \( f(x) = \frac{1}{x + 3} \) differ from \( f(x) = \frac{1}{x} \)?

What is the remainder when \( f(x) = 2x^3 – 3x^2 + 4x – 1 \) is divided by \( x = 2 \):

What is the approximate value of the \( y \)-intercept? The x-axis spans from below negative 5 to 5, and the y-axis spans from below negative 20 to above zero. The x-axis has a scale of 5 in increments of 1 and the y-axis has a scale of 10 in increments of 2. The red curve represents a polynomial function with four turning points. It begins from the bottom-middle of the third quadrant, reaches a local maximum at (negative 3, 0), then drops steeply to a local minimum near (negative 1.5, negative 21). It rises again to another local maximum near (0.5, 0.8), falls to a local minimum around (1, 0), and then steeply extends upward out of view.

For the function \( f(x) = \frac{1}{x^2 – 4} \), identify the vertical asymptotes.

Simplify the product: \[\frac{x^2 – 4x + 4}{x^2 – 9} \cdot \frac{x – 3}{x – 2}\]