Question

Find the value of the constant k that makes the function continuous. ​g(x)equals=startset start 2 by 2 matrix 1st row 1st column startfraction 2 x squared minus 4 x minus 16 over x minus 4 endfraction 2nd column if x not equals 4 2nd row 1st column kx minus 16 2nd column if x equals 4 endmatrix 2x2−4x−16 x−4 if x≠4 kx−16 if x=4

Answer 1

The value of k is 7.

Explanation:

Given function,

`f(x)=\left\{\begin{matrix}(2x^2-4x-16)/(x-4) & x\\eq 4\\ kx-16 & x=4\end{matrix}\right.`

The function f(x) would be continuous, if,

`\lim_(x\rightarrow 4) (2x^2-4x-16)/(x-4)=f(4)`

`\lim_(x\rightarrow 4) 2((x^2-2x-8)/(x-4))=4k-16`

`\lim_(x\rightarrow 4) 2((x^2-4x+2x-16)/(x-4))=4k-16`

`\lim_(x\rightarrow 4) 2((x(x-4)+2(x-4))/(x-4))=4k-16`

`\lim_(x\rightarrow 4) 2(((x+2)(x-4))/(x-4))=4k-16`

`\lim_(x\rightarrow 4) 2(x+2)=4k-16`

`2(4+2) = 4k-16`

`2(6)=4k-16`

`12+16 = 4k`

`\implies 4k=28\implies k=7`

Hence, the value of k would be 7.

Answer 2

You seem to have
.. g(x) = {(2x^2 -4x -16)/(x -4) . . . x ≠ 4
.. .. .. .. ..{ kx -16 . . . . . . . . . . . . . . x=4

The first expression can be simplified to
.. (2x^2 -4x -16)/(x -4) = 2(x +2)(x -4)/(x -4) = 2(x +2) . . . . x ≠ 4
At x=4, this simplified version has the value
.. 2(4 +2) = 12

To make the alternate definition of g(x) have that same value at x=4, we must have
.. k*4 -16 = 12
.. 4k = 28
,, k = 7

The constant k must be 7 for the function to be continuous at x=4.