What kind of function is , Let f:R→R be a function and f(x)=∣(x+4)(4x−10)

The function \( f: \mathbb{R} \to \mathbb{R} \) given by \( f(x) = |(x+4)(4x-10)| \) is an **absolute value function** of a quadratic expression. Let's analyze it in detail.

### Step 1: Expand the Quadratic Expression

Before applying the absolute value, expand the product:

\[
f(x) = |(x+4)(4x-10)|
\]

Using the distributive property (FOIL):

\[
(x+4)(4x-10) = x(4x) + x(-10) + 4(4x) + 4(-10)
\]
\[
= 4x^2 - 10x + 16x - 40
\]
\[
= 4x^2 + 6x - 40
\]

So, the function becomes:

\[
f(x) = |4x^2 + 6x - 40|
\]

### Step 2: Nature of the Function

This function is an **absolute value of a quadratic function**. Without the absolute value, the expression \( 4x^2 + 6x - 40 \) would be a standard quadratic function, which is a parabola. However, because of the absolute value, the function will reflect any negative parts of the quadratic upward.

1. The expression \( 4x^2 + 6x - 40 \) is a quadratic function, which is a parabola opening upwards since the coefficient of \( x^2 \) (which is 4) is positive.
   
2. The absolute value reflects any negative output of the quadratic to positive values, so the graph will have "V" shapes where the quadratic would otherwise dip below the x-axis.

### Step 3: General Behavior

- For values of \( x \) where \( 4x^2 + 6x - 40 \geq 0 \), the function will behave like the quadratic \( 4x^2 + 6x - 40 \).
- For values of \( x \) where \( 4x^2 + 6x - 40 < 0 \), the function will behave like the reflection of the quadratic above the x-axis, \( -(4x^2 + 6x - 40) \).

### Step 4: Function Type

The function \( f(x) = |(x+4)(4x-10)| \) is:

- **Piecewise-defined**: It changes form depending on whether \( 4x^2 + 6x - 40 \) is positive or negative.
- **Non-negative**: Because of the absolute value, the function always produces non-negative outputs.
- **Continuous**: The function is continuous everywhere on \( \mathbb{R} \).
- **Even symmetry**: It is not an even or odd function in general because the quadratic is not symmetric about the origin.

Would you like more details on how the graph of this function looks or about any specific property?

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Here are 5 related questions to deepen your understanding:
1. How can we find the x-intercepts of the quadratic function inside the absolute value?
2. What are the turning points of the quadratic function \( f(x) = 4x^2 + 6x - 40 \)?
3. How does the graph of \( f(x) = |4x^2 + 6x - 40| \) compare to the graph of \( g(x) = 4x^2 + 6x - 40 \)?
4. What happens to the behavior of the function at very large values of \( x \) (as \( x \to \infty \) or \( x \to -\infty \))?
5. How does changing the coefficients of a quadratic inside an absolute value affect the function's behavior?

**Tip**: When analyzing an absolute value function, it's helpful to first study the behavior of the function inside the absolute value, then apply the absolute value transformation to understand how the graph changes.

What kind of function is , Let f:R→R be a function and f(x)=∣(x+4)(4x−10)∣

This problem explores the function f(x) = |(x+4)(4x-10)|, an absolute value applied to a quadratic expression. Learn how to expand the quadratic, analyze the behavior of the function, and understand its properties. Suitable for high school students in Grades 9-12.

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