- COMMON ALGEBRA ERRORS–
975.log 5 (5x2) = 2log 5 (5x) = 2[log 5 (5) + log 5 (x)] =
2[1 + log 5 (x)]
a.The first equality is incorrect because 2log 5 (5x)
= log 5 (5x^2 ) = log 5 (25x^2 ). The other equalities
are correct.
b.The very last equality is incorrect because
log 5 5 = 0. The other equalities are correct.
c.There is no error.
976.ln(4x^2 – 1) = ln[(2x– 1)(2x+ 1)] + ln(2x– 1) +
ln(2x+ 1)
a.The “natural logarithm of a difference rule”
was not applied correctly. The correct state-
ment should be ln(4x^2 – 1) = ln(4x^2 ) – ln(1)
= ln(4x^2 ) – 0 =ln(4x^2 ). The last expression in
this string of equalities cannot be simplified
because the exponent 2 does not apply to the
entire input of the logarithm.
b.Using the fact that the natural logarithm of a
difference is the quotient of the natural loga-
rithms, we see that the expression ln(4x^2 – 1)
= lnl(n^41 x
^2 )
= ln( 04 x
^2 )
, so the expression is not
well–defined.
c.There is no error.
Set 63 (Answers begin on page 275.)
This problem set highlights common errors made in
graphing, computing with, and interpreting functions.
977.The vertical asymptote for the graph of
f(x) = xx 2 ++^24 is y= 0.
a.The expression should be factored and sim-
plified to obtain f(x) =x–^1 2. Then, we can
conclude that the vertical asymptote for fis
x= 2.
b.The line y= 0 is the horizontal asymptote for f.
c.There is no error.
978.The line x= ahas a slope of zero, for any real
number a.
a.The line is vertical, so its slope is undefined.
b.The statement is true except when a= 0.
The y–axis cannot be described by such
an equation.
c.There is no error.979.The point (–2, 1) lies in Quadrant IV.
a.The point is actually in Quadrant II.
b.The point is actually in Quadrant III.
c.There is no error.980.The inverse of the function f(x) = x^2 ,where xis
any real number, is the function f–1(x) = x.
a.fcannot have an inverse because it doesn’t
pass the vertical line test.
b.The domain offmust be restricted to [0,∞)
in order for fto have an inverse. In such case,
the given function f–1(x) = x is indeed its
inverse.
c.There is no error.981.The lines y= 3x+ 2 and y= –^13 x+ 2 are
perpendicular.
a.The lines are parallel since their slopes are
negative reciprocals of each other.
b.The lines cannot be perpendicular since the
product of their slopes is not 1.
c.There is no error.982.The slope of a line passing through the points
(a, b) and (c, d) is m=ba––dc, provided that a≠c.
a.The slope is actually equal to the quantity m
=ba––dc, provided that b≠d.
b.The slope is actually equal to the quantity m
=bd––ac, provided that c≠d.
c.There is no error.