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QuestionFind the value of mm for the continuous function f(x)={mx−8 if x<−7x2+10x−1 if x≥−7f(x)=\left\{\begin{array}{ll}m x-8 & \text { if } \quad x<-7 \\ x^{2}+10 x-1 & \text { if } \quad x \geq-7\end{array}\right.

Studdy Solution

STEP 1

Assumptions1. The function f(x)f(x) is given by f(x)={mx−8 if x<−7x+10x−1 if x≥−7f(x)=\left\{\begin{array}{ll}m x-8 & \text { if } \quad x<-7 \\ x^{}+10 x-1 & \text { if } \quad x \geq-7\end{array}\right.. The function f(x)f(x) is continuous everywhere.

STEP 2

For a piecewise function to be continuous at a point where the function definition changes (in this case, x=−7x=-7), the two pieces of the function must meet at that point. This means that the value of the function at x=−7x=-7 must be the same whether we use the first piece (mx−8mx-8) or the second piece (x2+10x−1x^{2}+10x-1).
So, we can set up the equationm(−7)−8=(−7)2+10(−7)−1m(-7) -8 = (-7)^2 +10(-7) -1

STEP 3

Now, we simplify the equation to solve for mm.
−7m−8=49−70−1-7m -8 =49 -70 -1

STEP 4

implify the right side of the equation.
−7m−8=−22-7m -8 = -22

STEP 5

Add8 to both sides of the equation to isolate the term with mm on one side.
−7m=−22+8-7m = -22 +8

STEP 6

implify the right side of the equation.
−m=−14-m = -14

STEP 7

Finally, divide both sides of the equation by -7 to solve for mm.
m=−14/−7m = -14 / -7

STEP 8

implify the right side of the equation to find the value of mm.
m=2m =2So, for the function f(x)f(x) to be continuous everywhere, mm must be equal to2.

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