Let's solve the problem from the image you uploaded.

Given:

• $f(x) = x^2 - 3x + 4$
• $(f + g)(x) = 3x^2 + x - 5$

To Find:

a. $g(x)$
b. $f(3)$ and $g(3)$
c. $(f + g)(3)$ and $(f - g)(3)$

a. Finding $g(x)$:

Since $(f + g)(x) = f(x) + g(x)$, we can write: $g(x) = (f + g)(x) - f(x)$

Substituting the given functions: $g(x) = \left(3x^2 + x - 5\right) - \left(x^2 - 3x + 4\right)$ Simplifying the expression: $g(x) = 3x^2 + x - 5 - x^2 + 3x - 4$ $g(x) = (3x^2 - x^2) + (x + 3x) + (-5 - 4)$ $g(x) = 2x^2 + 4x - 9$

So, $g(x) = 2x^2 + 4x - 9$.

b. Finding $f(3)$ and $g(3)$:

• $f(3)$: $f(3) = 3^2 - 3(3) + 4 = 9 - 9 + 4 = 4$

• $g(3)$: $g(3) = 2(3^2) + 4(3) - 9 = 2(9) + 12 - 9 = 18 + 12 - 9 = 21$

So, $f(3) = 4$ and $g(3) = 21$.

c. Finding $(f + g)(3)$ and $(f - g)(3)$:

• $(f + g)(3)$: $(f + g)(3) = f(3) + g(3) = 4 + 21 = 25$

• $(f - g)(3)$: $(f - g)(3) = f(3) - g(3) = 4 - 21 = -17$

So, $(f + g)(3) = 25$ and $(f - g)(3) = -17$.

Do you want more details or have any questions?

Related Questions:

1. How do you find the difference of two functions?
2. What happens when you subtract a quadratic function from another quadratic function?
3. How does the value of $g(x)$ change when you modify $f(x)$ or $(f + g)(x)$?
4. Can you explain how to factor a quadratic function like $g(x)$?
5. What would happen if the coefficient of $x^2$ in $(f + g)(x)$ were different?

Tip: Always substitute carefully when working with algebraic expressions, especially when subtracting functions.