It seems you have three functions, $F(x) = x + 3$, $G(x) = 2x + 7$, and $H(x) = x^2$, and you want to know how to work with them. There are several ways to manipulate and combine these functions, depending on what you want to achieve. Here are some of the basic operations you can perform:

1. Function Evaluation

You can plug a specific value of $x$ into each function to calculate its output. For example:

• For $F(x) = x + 3$, if $x = 2$: $F(2) = 2 + 3 = 5$

• For $G(x) = 2x + 7$, if $x = 2$: $G(2) = 2 \cdot 2 + 7 = 4 + 7 = 11$

• For $H(x) = x^2$, if $x = 2$: $H(2) = 2^2 = 4$

2. Function Addition

You can add two functions together to create a new function:

$(F + G)(x) = F(x) + G(x)$

For the given functions:

$(F + G)(x) = (x + 3) + (2x + 7) = 3x + 10$

3. Function Composition

You can compose two functions, which means applying one function to the result of another:

$(F \circ G)(x) = F(G(x))$

For example:

$(F \circ G)(x) = F(2x + 7) = (2x + 7) + 3 = 2x + 10$

Similarly:

$(G \circ H)(x) = G(H(x)) = G(x^2) = 2(x^2) + 7 = 2x^2 + 7$

4. Function Multiplication

You can also multiply two functions together:

$(F \cdot G)(x) = F(x) \cdot G(x)$

For the given functions:

$(F \cdot G)(x) = (x + 3)(2x + 7)$

To simplify, use the distributive property:

$(F \cdot G)(x) = 2x^2 + 7x + 6x + 21 = 2x^2 + 13x + 21$

5. Finding Inverses (if possible)

You can find the inverse of a function, which "reverses"