Math

QuestionFind the value of bb that makes the piecewise function f(x)f(x) continuous at x=9x=9: f(x)={2x5 if x9,3x+b if x>9}f(x)=\{2x-5 \text{ if } x \leq 9, -3x+b \text{ if } x>9\}.

Studdy Solution

STEP 1

Assumptions1. The function f(x)f(x) is defined as a piecewise function x5x -5 if x9x \leq9 and 3x+b-3x + b if x>9x >9. . 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=9x =9), the value of the function from the left side of that point must equal the value of the function from the right side of that point.
So, we have to set the two parts of the function equal to each other at x=9x =9.
2x5=x+batx=92x -5 = -x + b \quad \text{at} \quad x =9

STEP 3

Substitute x=9x =9 into the equation.
2(9)5=3(9)+b2(9) -5 = -3(9) + b

STEP 4

Calculate the left and right sides of the equation.
18=27+b18 - = -27 + b

STEP 5

implify the equation.
13=27+b13 = -27 + b

STEP 6

To solve for bb, we need to isolate bb on one side of the equation. We can do this by adding 2727 to both sides of the equation.
13+27=27+27+b13 +27 = -27 +27 + b

STEP 7

implify the equation to find the value of bb.
40=b40 = bSo, for the function f(x)f(x) to be continuous everywhere, we must have b=40b =40.

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