If f(x) = 2x2 − x + 4, find the following.f(2) = f(−2)= f(a)= f(−a)= f(a + 1)= 2f(a)= f(2a)= f(a2)= [f(a)]2= f(a + h)=

1. Break Down the Problem

We need to evaluate the function $f(x) = 2x^2 - x + 4$ for various inputs, including specific values and expressions involving $a$ and $h$.

2. Relevant Concepts

To evaluate $f$ at different arguments, we simply substitute the values into the expression for $f(x)$.

3. Analysis and Detail

Let's compute each requirement one by one.

1. For $f(2)$:
   $f(2) = 2(2)^2 - (2) + 4 = 2 \cdot 4 - 2 + 4 = 8 - 2 + 4 = 10$

2. For $f(-2)$:
   $f(-2) = 2(-2)^2 - (-2) + 4 = 2 \cdot 4 + 2 + 4 = 8 + 2 + 4 = 14$

3. For $f(a)$:
   $f(a) = 2(a)^2 - (a) + 4 = 2a^2 - a + 4$

4. For $f(-a)$:
   $f(-a) = 2(-a)^2 - (-a) + 4 = 2a^2 + a + 4$

5. For $f(a + 1)$:
   $f(a + 1) = 2(a + 1)^2 - (a + 1) + 4$
   First, expand $(a + 1)^2$:
   $(a + 1)^2 = a^2 + 2a + 1$
   Substituting:
   $f(a + 1) = 2(a^2 + 2a + 1) - (a + 1) + 4 = 2a^2 + 4a + 2 - a - 1 + 4 = 2a^2 + 3a + 5$

6. For $2f(a)$:
   $2f(a) = 2(2a^2 - a + 4) = 4a^2 - 2a + 8$

7. For $f(2a)$:
   $f(2a) = 2(2a)^2 - (2a) + 4 = 2 \cdot 4a^2 - 2a + 4 = 8a^2 - 2a + 4$

8. For $f(a^2)$:
   $f(a^2) = 2(a^2)^2 - (a^2) + 4 = 2a^4 - a^2 + 4$

9. For $[f(a)]^2$:
   $[f(a)]^2 = (2a^2 - a + 4)^2$
   Expanding using $(x + y + z)^2$:
   $(2a^2)^2 + (-a)^2 + 4^2 + 2(2a^2)(-a) + 2(2a^2)(4) + 2(-a)(4)$
   $= 4a^4 + a^2 + 16 - 4a^3 + 16a^2 - 8a = 4a^4 - 4a^3 + 17a^2 - 8a + 16$

10. For $f(a + h)$:
    $f(a + h) = 2(a + h)^2 - (a + h) + 4$
    Expanding:
    $(a + h)^2 = a^2 + 2ah + h^2$ Thus,
    $f(a + h) = 2(a^2 + 2ah + h^2) - (a + h) + 4 = 2a^2 + 4ah + 2h^2 - a - h + 4$ Simplifying:
    $= 2a^2 + 4ah + 2h^2 - a - h + 4$

4. Verify and Summarize

The computations have been carried out step-by-step for each condition provided. Each expression has been simplified properly.

Final Answer

• $f(2) = 10$
• $f(-2) = 14$
• $f(a) = 2a^2 - a + 4$
• $f(-a) = 2a^2 + a + 4$
• $f(a + 1) = 2a^2 + 3a + 5$
• $2f(a) = 4a^2 - 2a + 8$
• $f(2a) = 8a^2 - 2a + 4$
• $f(a^2) = 2a^4 - a^2 + 4$
• $[f(a)]^2 = 4a^4 - 4a^3 + 17a^2 - 8a + 16$
• $f(a + h) = 2a^2 + 4ah + 2h^2 - a - h + 4$