Can someone help me with these two continuous function problems? Information is in the pictures.

Jan 20, 2018

$c = 1$

$b = 18$

Explanation:

For the first question we need to find the value of $c$ such that:

${\lim}_{x \to {c}^{-}} {x}^{2} - 7 = {\lim}_{x \to {c}^{+}} 2 x - 8$

Since $f \left(x\right)$ is expected to be the same approaching $c$ from either side (by the definition of continuous) we can evaluate both limits by direct substitution:

${\lim}_{x \to {c}^{-}} {x}^{2} - 7 = {\lim}_{x \to {c}^{+}} 2 x - 8$

$\to {c}^{2} - 7 = 2 c - 8$

Now solve for $c$:

${c}^{2} - 2 c + 1 = 0$

${\left(c - 1\right)}^{2} = 0 \therefore c = 1$

From the image it can be seen the graph is continuous when $c = 1$. If we were to choose a value other than $c = 1$ then we would get a discontinuity. Say, for example, $c = 0$:

The graph is now clearly broken at $x = 0$ as the piecewise condition creates a jump in the value.

For the second part follow the same procedure:

${\lim}_{x \to {2}^{-}} 3 x - 4 = {\lim}_{x \to {2}^{+}} - 8 x + b$

$3 \left(2\right) - 4 = - 8 \left(2\right) + b$

$6 - 4 = - 16 + b$

$\to b = 2 + 16 = 18$

Jan 20, 2018

$c = 1$
$b = 18$

Explanation:

to find out where one graph ends and another graph starts, you can use the point of intersection.

to find the $x -$coordinate for the point of intersection, find the value of $x$ for which $y$ is equal for both functions.

${x}^{2} - 7 = 2 x - 8$

then solve for $x$:

${x}^{2} - 2 x - 7 = - 8$

${x}^{2} - 2 x + 1 = 0$

$\left(x - 1\right) \left(x - 1\right) = 0$

$x - 1 = 0$

$x = 1$ (repeated root)

the $y -$coordinate is ${1}^{2} - 7 = 2 - 8 = - 6$

hence, the point of intersection is

when $x \le 1 , f \left(x\right) = {x}^{2} - 7$
when $x > 1 , f \left(x\right) = 2 x - 8$

$c = 1$

this is the graph:

-

the inequalities $\left\{x \le 2\right\}$ and $x > 2$ show that the $x -$value of the point of intersection is $2$.

at $x = 2$, $3 x - 4 = 6 - 4 = 2$

this means that $y = 2$ at the point of intersection

therefore at $x = 2 , - 8 x + b = 2$

$2$ can be substituted for $x$ in solving for $b$:

$- 8 x + b = 2$

$b = 2 + 8 x$

$b = 2 + 16$

$b = 18$

the point of intersection is $\left(2 , 2\right)$

to draw the graph:

draw the graph of $3 x - 4$ up to the point $\left(2 , 2\right)$
from there, draw the graph $- 8 x + 18$

this is a digital representation: