𝑓(𝑥)=3𝑥+5−2𝑥2
 , evaluate: f(x+h) =f(x-h) =

Given the function \( f(x) = 3x + 5 - 2x^2 \), we need to evaluate \( f(x+h) \) and \( f(x-h) \).

1. **Evaluate \( f(x+h) \):**

   Substitute \( x+h \) into the function:
   \[
   f(x+h) = 3(x+h) + 5 - 2(x+h)^2
   \]
   Simplify this:
   \[
   f(x+h) = 3x + 3h + 5 - 2(x^2 + 2xh + h^2)
   \]
   Distribute and combine like terms:
   \[
   f(x+h) = 3x + 3h + 5 - 2x^2 - 4xh - 2h^2
   \]
   So, the expression becomes:
   \[
   f(x+h) = -2x^2 - 4xh - 2h^2 + 3x + 3h + 5
   \]

2. **Evaluate \( f(x-h) \):**

   Substitute \( x-h \) into the function:
   \[
   f(x-h) = 3(x-h) + 5 - 2(x-h)^2
   \]
   Simplify this:
   \[
   f(x-h) = 3x - 3h + 5 - 2(x^2 - 2xh + h^2)
   \]
   Distribute and combine like terms:
   \[
   f(x-h) = 3x - 3h + 5 - 2x^2 + 4xh - 2h^2
   \]
   So, the expression becomes:
   \[
   f(x-h) = -2x^2 + 4xh - 2h^2 + 3x - 3h + 5
   \]

Thus, the expressions for \( f(x+h) \) and \( f(x-h) \) are:
\[
f(x+h) = -2x^2 - 4xh - 2h^2 + 3x + 3h + 5
\]
\[
f(x-h) = -2x^2 + 4xh - 2h^2 + 3x - 3h + 5
\]

Would you like more details or have any questions?

Here are five related questions you might find interesting:
1. How does the function \( f(x) \) behave as \( h \) approaches zero?
2. What is the difference \( f(x+h) - f(x-h) \) in terms of \( h \)?
3. What is the derivative of \( f(x) \) with respect to \( x \)?
4. How would the expressions change if the function had a different quadratic coefficient?
5. How can we interpret the symmetry or asymmetry of \( f(x+h) \) and \( f(x-h) \)?

**Tip:** When evaluating \( f(x+h) \) and \( f(x-h) \), it's helpful to expand and simplify each term carefully to avoid mistakes.

f(x) = 3x + 5 - 2x^2, evaluate: f(x+h) =f(x-h) =

Explore how to evaluate the function f(x) = 3x + 5 - 2x^2 by finding expressions for f(x+h) and f(x-h). This detailed guide covers algebraic techniques including function substitution and quadratic expansion, suitable for students in Grades 9-12.

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