Math Snap

STEP 1

Assumptions1. We have two functions, $f(x)$ and $g(x)$, which are defined differently for different ranges of $x$.
. The function $f(x)$ is defined as $5x+4$ for $x \geq$ and as $x^+4x$ otherwise.
3. The function $g(x)$ is defined as $-3x+1$ for $x \leq0$ and as $x-7$ for $x >0$.
4. We need to find the sum function $(f+g)(x)$.

STEP 2

The sum function $(f+g)(x)$ is defined as $f(x) + g(x)$. Since both $f(x)$ and $g(x)$ are defined differently for different ranges of $x$, we need to consider these ranges separately.

STEP 3

Let's first consider the range $x \leq0$. In this range, $f(x) = x^2+x$ and $g(x) = -3x+1$. So, the sum function $(f+g)(x)$ is$$(f+g)(x) = f(x) + g(x) = x^2+x + (-3x+1)$$

STEP 4

implify the expression for $(f+g)(x)$ for $x \leq0$.
$$(f+g)(x) = x^2+4x -3x +1 = x^2+x+1$$

STEP 5

Next, consider the range $0 < x <2$. In this range, $f(x) = x^2+4x$ and $g(x) = x-7$. So, the sum function $(f+g)(x)$ is$$(f+g)(x) = f(x) + g(x) = x^2+4x + (x-7)$$

STEP 6

implify the expression for $(f+g)(x)$ for $0 < x <2$.
$$(f+g)(x) = x^2+4x + x - = x^2+5x-$$

STEP 7

Finally, consider the range $x \geq2$. In this range, $f(x) =5x+4$ and $g(x) = x-7$. So, the sum function $(f+g)(x)$ is$$(f+g)(x) = f(x) + g(x) =5x+4 + (x-7)$$

STEP 8

implify the expression for $(f+g)(x)$ for $x \geq2$.
$$(f+g)(x) =5x+4 + x -7 =6x-3$$

Now we have the sum function $(f+g)(x)$ for all possible ranges of $x$. So, we can write$$(f+g)(x) = \left\{\begin{array}{c}x^2+x+ \text { if } x \leq \\ x^2+5x-7 \text { if } < x <2 \\6x-3 \text { if } x \geq2\end{array}\right.$$ This is the sum function $(f+g)(x)$.