To analyze the function \( f(-x) = 8x^2 + 4x - 5 \), we can find the original function \( f(x) \) by understanding how the transformation \( f(-x) \) relates to \( f(x) \). The function \( f(-x) \) replaces \( x \) with \( -x \) in \( f(x) \). So to find \( f(x) \), we can reverse the transformation by substituting \( -x \) back with \( x \).

Given:
\[
f(-x) = 8x^2 + 4x - 5
\]
Since \( x^2 \) remains the same whether \( x \) is positive or negative, the \( 8x^2 \) term stays unchanged. However, the linear term \( 4x \) will change sign when we substitute \( -x \) for \( x \).

Thus, the function \( f(x) \) is:
\[
f(x) = 8x^2 - 4x - 5
\]

Would you like a detailed explanation on how transformations like \( f(-x) \) affect functions?

### Related Questions:
1. What are the effects of applying transformations like \( f(x+c) \) or \( f(x-c) \) on a function?
2. How do you determine if a function is even or odd based on \( f(-x) \)?
3. Can you explain the difference between horizontal and vertical shifts in functions?
4. How do you find the inverse of a quadratic function?
5. What is the geometric interpretation of \( f(-x) \) in terms of reflection?

### Tip:
When working with transformations, always check whether they affect the signs or powers of \( x \), as this determines how the graph changes.

Explore how to analyze and transform the function f(-x) = 8x^2 + 4x - 5 into f(x). This guide covers key algebraic concepts, including function transformation, symmetry in functions, and properties of quadratic equations. Ideal for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation