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Esteban MontielStudios

Answered question

2022-04-08

Find the constant a such that the function is continuous on the entire real number line.

$f (x) = {\begin{array}{ll} x^{3} & x \leq 9 \\ a x^{2} & x > 9 \end{array}$

Vasquez

Expert2023-04-27Added 669 answers

We want the function

f (x)

to be continuous on the entire real number line. This means that the left and right limits of

f (x)

x = 9

must exist and be equal, and the value of

f (x)

x = 9

must be the same as the limit.
First, let's find the left limit of

f (x)

x = 9

{lim}_{x \to 9^{-}} f (x) = {lim}_{x \to 9^{-}} x^{3} = 9^{3} = 729

Next, let's find the right limit of

f (x)

x = 9

{lim}_{x \to 9^{+}} f (x) = {lim}_{x \to 9^{+}} a x^{2} = a \times 9^{2} = 81 a

Since

f (x)

is continuous at

x = 9

, the left and right limits must be equal:

{lim}_{x \to 9^{-}} f (x) = {lim}_{x \to 9^{+}} f (x)

729 = 81 a

Solving for

a

, we get:

a = \frac{729}{81} = 9

Therefore, the constant

a

that makes

f (x)

continuous on the entire real number line is

a = 9

. So the function becomes:

f (x) = {\begin{matrix} x^{3} & x \leq 9 \\ 9 x^{2} & x > 9 \end{matrix}

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