Math Snap

STEP 1

Assumptions1. The function $f(x)$ is defined as a piecewise function with two parts $mx-7$ for $x<-4$ and $x^{}+5x-3$ for $x \geq -4$.
. The function $f(x)$ is continuous everywhere, which means the two parts of the function must connect at $x=-4$.

STEP 2

To ensure the function is continuous at $x=-4$, the values of $f(x)$ for both parts of the function must be equal at $x=-4$. Therefore, we can set up the following equation$$mx-7 = x^{2}+5x-$$where $x=-4$.

STEP 3

Substitute $x=-$ into the equation.
$$m(-)-7 = (-)^{2}+5(-)-3$$

STEP 4

implify the equation.
$$-4m-7 =16-20-3$$

STEP 5

Further simplify the equation.
$$-4m-7 = -7$$

STEP 6

Now, we can solve for $m$. First, add to both sides of the equation to isolate $-4m$ on one side.
$$-4m =0$$

Finally, divide both sides of the equation by -4 to solve for $m$.
$$m =0$$So, for the function $f(x)$ to be continuous everywhere, $m$ must be equal to0.