Resolved: "g(x) =2x2−3x−20x−4,ifx≠4 and =kx-15, if x=4 Evaluate the..."

College
Algebra
College algebra
Transformations of functions

UkusakazaL

Answered question

2021-09-17

g(x)

= \frac{2 x^{2} - 3 x - 20}{x - 4}, if x \neq 4

and
=kx-15, if x=4
Evaluate the constant k that makes the function continuous.

Answer & Explanation

Arnold Odonnell

Skilled2021-09-18Added 109 answers

For f(x) to be continuous at a point x=a:

lim_{x \to a^{+}} f (x) = lim_{x \to a^{-}} f (x) = f (a)

It's given that g(x) is continuous at x=4, so

lim_{x \to 4^{+}} g (x) = lim_{x \to 4^{-}} g (x) = g (4)

lim_{x \to 4} g (x) = g (4)

lim_{x \to 4} \frac{2 x^{2} - 3 x - 20}{x - 4} = k (4) - 15

lim_{x \to 4} \frac{2 x^{2} - 8 x + 5 x - 20}{x - 4} = 4 k - 15

lim_{x \to 4} \frac{2 x (x - 4) + 5 (x - 4)}{x - 4} = 4 k - 15

lim_{x \to 4} \frac{(2 x + 5) (x - 4)}{x - 4} = 4 k - 15

lim_{x \to 4} (2 x + 5) = 4 k - 15

2(4)+5=4k-15
k=7

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