QuestionFind the value of $m$ for the continuous piecewise function $f(x)=\left\{\begin{array}{ll}m x-7 & \text { if } \quad x<-4 \\ x^{2}+5 x-3 & \text { if } \quad x \geq-4\end{array}\right.$

Studdy Solution

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.
$-4m =0$

STEP 7

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.

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QuestionFind the value of $m$ for the continuous piecewise function $f(x)=\left\{\begin{array}{ll}m x-7 & \text { if } \quad x<-4 \\ x^{2}+5 x-3 & \text { if } \quad x \geq-4\end{array}\right.$

Studdy Solution

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.
$-4m =0$

STEP 7

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.

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QuestionFind the value of mmm for the continuous piecewise function f(x)={mx−7 if x<−4x2+5x−3 if x≥−4f(x)=\left\{\begin{array}{ll}m x-7 & \text { if } \quad x<-4 \\ x^{2}+5 x-3 & \text { if } \quad x \geq-4\end{array}\right.f(x)={mx−7x2+5x−3​ if x<−4 if x≥−4​

Studdy Solution

STEP 1

STEP 2

To ensure the function is continuous at x=−4x=-4x=−4, the values of f(x)f(x)f(x) for both parts of the function must be equal at x=−4x=-4x=−4. Therefore, we can set up the following equationmx−7=x2+5x−mx-7 = x^{2}+5x-mx−7=x2+5x−where x=−4x=-4x=−4.

STEP 3

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

STEP 4

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

STEP 5

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

STEP 6

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

STEP 7

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

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QuestionFind the value of $m$ for the continuous piecewise function $f(x)=\left\{\begin{array}{ll}m x-7 & \text { if } \quad x<-4 \\ x^{2}+5 x-3 & \text { if } \quad x \geq-4\end{array}\right.$

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 $x=-4$ .

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

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

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

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.