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

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2 Answers
Jan 20, 2018

#c=1#

#b=18#

Explanation:

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

#lim_(x->c^-)x^2-7=lim_(x->c^+)2x-8#

Since #f(x)# 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->c^-)x^2-7=lim_(x->c^+)2x-8#

#->c^2-7=2c-8#

Now solve for #c#:

#c^2-2c+1=0#

#(c-1)^2=0 therefore c=1#

Generated on Mathematica

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#:

Generated on Mathematica

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->2^-)3x-4=lim_(x->2^+)-8x+b#

#3(2)-4=-8(2)+b#

#6-4=-16+b#

#-> 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 = 2x - 8#

then solve for #x#:

#x^2 -2x - 7 = -8#

#x^2 - 2x + 1 = 0#

#(x-1)(x-1) = 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<=1, f(x) = x^2 - 7#
when #x>1, f(x) = 2x - 8#

#c = 1#

this is the graph:

desmos.com/calculator

-

the inequalities #{x<=2}# and #x>2# show that the #x-#value of the point of intersection is #2#.

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

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

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

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

#-8x + b = 2#

#b = 2 + 8x#

#b = 2 + 16#

#b = 18#

the point of intersection is #(2,2)#

to draw the graph:

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

this is a digital representation:

desmos.com/calculator