- b.Observe that 6x– 13 ^4 x+ 6 = 0 can be
written as 6(^4 x)^2 – 13(^4 x) + 6 = 0. Let u =
^4 x.Rewriting the original equation yields
the equation 6u^2 – 13u + 6 = 0, which is qua-
dratic. Factoring yields the equivalent equa-
tion (2u – 3)(3u – 2) = 0. Solving this equation
foruyields the solutions u = ^23 or u= ^32 . Solv-
ing the original equation requires that we go
back to the substitution and write u in terms
of the original variable x:
u= ^23 is the same as ^4 x= ^23 , so that x= (^23 )^4 = ^1861 .
u= ^32 is the same as ^4 x= ^32 , so that x= ( 23 )^4 = ^8116 .
Therefore, the solution of the original equation
is x= ^1861 ,^8116 .
- c.Let u = a. Observe that 2a – 11a + 12 = 0
can be written as 2(a )^2 – 11(a ) + 12 = 0.
Rewriting the original equation yields the
equation 2u^2 – 11u + 12 = 0, which is qua-
dratic. Factoring yields the equivalent equa-
tion (2u – 3)(u– 4) = 0. Solving this equation
for uyields the solutions u = ^32 or u= 4. In
order to solve the original equation, we go
back to the substitution and write u in terms
of the original variable a:
u = ^32 is the same as a = ^32 , so x= (^32 )^3 = ^287
u= 4 is the same as a = 4, so x= (4)^3 = 64
The solutions of the original equation are
a= 64,^287 .
Section 6—Elementary
Functions
Set 43 (Page 102)
- b.Draw a horizontal line across the coordinate
plane where f(x) = 3. This line touches the
graph off(x) in exactly one place. Therefore,
there is one value for which f(x) = 3.
- d.The x-axis is the graph of the line f(x) = 0,
so every time the graph touches the x-axis.
The graph off(x) touches the x-axis in 5 places.
Therefore, there are 5 values for which f(x) = 0.
- b.Draw a horizontal line across the coordinate
plane where f(x) = 10. The arrowheads on the
ends of the curve imply that the graph extends
upward, without bound, as x tends toward
both positive and negative infinity. This line
touches the graph off(x) in 2 places. There-
fore, there are 2 values for which f(x) = 10.
- e.The domain of a real-valued function is
the set of all values that, when substituted
for the variable, produce a meaningful output,
while the range of a function is the set of all
possible outputs. All real numbers can be
substituted for xin the function f(x) = x^2 – 4,
so the domain of the function is the set of all
real numbers. Since the xterm is squared, the
smallest value that this term can equal is 0
(when x= 0). Therefore, the smallest value that
f(x) can attain occurs when x= 0. Observe that
f(0) = 0^2 – 4 = –4. The range off(x) is the set of
all real numbers greater than or equal to –4.
^1
3
^1
3
^1
3
^1
3
^1
3
^2
3
^1
3
ANSWERS & EXPLANATIONS–
