Section 6.1: Composite Functions

• Def: Given two function f and g, the composite function, which we denoteby f ◦ g and read as “f composed with g,” is defined by (f ◦ g)(x) = f(g(x)).In other words, the function f composed with g is the function you get byputting the function g into the function f . The domain of f ◦ g is the set ofall x in the domain of g such that g(x) is in the domain of f .

• ex. Given f(x) =√

x + 1 and g(x) = 3x, find:

a) (f ◦ g)(8)

b) (g ◦ f)(3)

c) (f ◦ f)(15)

d) (g ◦ g)(1)

• ex. For the given function, find: a)f ◦ g, b)g ◦ f , c)f ◦ f , d)g ◦ g, and statethe domain of each:

1

i) f(x) = 1x+3

, g(x) = −2x

ii) f(x) = x2 + 4, g(x) =√

x− 2

• ex. Find functions f and g such that f ◦ g = H.

a) H(x) = (3x2 + 2x− 1)4

b) H(x) =√

4x2 − 9

c) H(x) = |3x− 5|

2

Section 6.2: One-to-One Functions; Inverse Functions

• Def: Suppose that f is a function. The inverse of f is the correspondencewhich takes f(x) as the input and gives back x as the output. The domainof f is the range of the inverse of f and the range of f is the domain of theinverse of f .

• ex. Find the inverse of the following functions and determine whether theinverse is a function.

a)

b)

c) {(1, 2), (2, 8), (3, 18), (4, 32)}

• Def: When the inverse of a function is itself a function, then we call f a one-to-one function. In other words, f is a one-to-one function if, for any choiceof elements x1 and x2 in the domain of f , with x1 6= x2, the correspondingvalues f(x1) and f(x2) are not equal in the range of f .

• Horizontal-line Test: If every possible horizontal line that you can draw inthe xy-plane intersects the graph of a function f in at most one point, thenf is one-to-one.

• ex. Determine whether the following functions are one-to-one or not:

1

a) f(x) = (x− 1)2

b) g(x) = x3 + 2

2

c)

• Notation: The inverse of f is denoted by f−1. Note that this does not mean1

f(x).

• Facts:

i) Domain of f = Range of f−1 and Range of f = Domain of f−1 (we statedthis in the definition of the inverse of f , but now we can write it usingour notation of f−1).

ii) f(f−1(x)) = x and f−1(f(x)) = x

• ex. Verify that the functions f and g are inverses of each other.

a) f(x) = 3x + 4; g(x) = 13(x− 4)

b) f(x) = (x + 3)2, x ≥ −3; g(x) =√

x− 3

3

c) f(x) = 2−x3+x

; g(x) = 2−3x1+x

• Theorem: The graph of a function f and the graph of its inverse f−1 aresymmetric with respect to the line y = x. Symmetry about the line y = xmeans the x- and y-coordinates are switched.

• ex. The graph of a one-to-one function f is given. Draw the graph of f−1.The graph of y = x is also already given.

a)

4

b)

• To find the inverse of the function y = f(x), interchange the x and y to getx = f(y). Then, if possible, solve for y. This will give you f−1.

• ex. The function f is one-to-one. Find its inverse. State the domain andrange of f and f−1.

a) f(x) = x3 + 1

b) f(x) = −3xx+1

5

c) f(x) = 3x+2x−6

d) f(x) = x2+33x2 , x > 0

6

Section 6.2: Example Answers

• ex. Determine whether the following functions are one-to-one or not:

a) f(x) = (x− 1)2

b) g(x) = x3 + 2

1

c)

• ex. The graph of a one-to-one function f is given. Draw the graph of f−1.The graph of y = x is also already given.

a)

2

b)

3

Section 6.3: Exponential Functions

• Def: An exponential function is a function of the form

f(x) = ax

where a is a positive real number and a 6= 1. The domain of f is the set ofall real numbers.

• Exponent Laws:

– am · an = am+n

– (am)n = amn

– (ab)n = an · bn– am/n = n

√am = ( n

√a)

m

–(ab

)n= an

bn

– a−n = 1an

– 1a−n = an

– a0 = 1

– 1n = 1

• Properties of the Exponential Function f(x) = ax, a > 1:

1. The domain is the set of all real numbers. The range is the set of positivereal numbers.

2. There are no x-intercepts and the y-intercept is 1.

3. The line y = 0 (the x-axis) is the horizontal asymptote as x→ −∞.

4. f is always increasing and is therefore one-to-one on (−∞,∞).

5. The graph of f contains the points (0, 1), (1, a), (−1, 1a).

6. Graphs of the exponential function f(x) = ax for a = 2, 3, 4.

1

• Properties of the Exponential Function f(x) = ax, 0 < a < 1:

1. The domain is the set of all real numbers. The range is the set of positivereal numbers.

2. There are no x-intercepts and the y-intercept is 1.

3. The line y = 0 (the x-axis) is the horizontal asymptote as x→∞.

4. f is always decreasing and is therefore one-to-one on (−∞,∞).

5. The graph of f contains the points (0, 1), (1, a), (−1, 1a).

6. Graphs of the exponential function f(x) = ax for a = 12, 13, 14.

• Just like we use π to symbolize the number π ≈ 3.14159, use use e to sym-bolize the number e ≈ 2.718281. The number e, just like π, is very importantin mathematics and comes up often in applications. We call the functionf(x) = ex the exponential function, even though any function of the formf(x) = ax, where a is any positive real number, is an exponential function.

• ex. Graph the function then state the domain, range, and horizontal asymp-tote:

2

a) f(x) = ex − 1

b) f(x) = 3ex+2

3

c) f(x) = 9− 2−x

• Fact: If au = av, then u = v.

• ex. Solve:

a) 51−2x = 15

b) 8x2−2x = 1

2

4

c) 4x2= 2x

d)(12

)x= 4

e) 4x − 2x = 0

• ex. If 2x = 3, what does 4−x equal?

5

Section 6.3: Example Answers

• ex. Graph the function then state the domain, range, and horizontal asymp-tote:

a) f(x) = ex − 1

b) f(x) = 3ex+2

1

c) f(x) = 9− 2−x

2

Section 6.4: Logarithmic Functions

• Def: The logarithmic function to the base a > 0, denoted by y = loga x andread as “log base a of x”, is the inverse function of the exponential functiony = ax. loga x is defined to be the exponent that a needs to have in orderto give you the value x. In other words, y = loga x is equivalent to writingx = ay. The domain of y = loga x is x > 0 and the range is (−∞, infty).

• Notation: We denote loge by ln and call it the natural logarithm. We de-note log10 by log (so if you just see log without any base specified, then itautomatically means base 10) and call it the common logarithm.

• ex. Change the exponential expression to an equivalent expression involvinga logarithm:

a) 64 = 82

b) 2x = 6.2

c) 2.23 = N

• ex. Change the logarithmic expression into an equivalent expression involvingan exponent:

a) log4

(164

)= −3

b) log3 8 = x

c) ln x = 2.1

• ex. Find the exact value:

1

a) log6 6

b) log43√

16

c) log√2 4

d) ln e4

• The logarithmic function is the inverse of the exponential function, so thedomain of the logarithmic function is the same as the range of the exponentialfunction, which is (0,∞), and the range of the logarithmic function is thesame as the domain of the exponential function, which is (−∞,∞).

• ex. Find the domain:

a) g(x) = ln(x− 4)

b) h(x) = log3

(x

x−3

)

2

• Since the logarithmic function is the inverse of the exponential function, thegraph of the logarithmic function is the graph of the exponential function,but reflected about the line y = x.

• Properties of the graph of a Logarithmic Function f(x) = loga x:

1. The domain is the set of all positive real numbers. The range is the setof all real numbers.

2. The x-intercept of the graph is 1. There is no y-intercept.

3. The vertical asymptote is the line x = 0 (the y-axis).

4. If 0 < a < 1 then the function is decreasing. If a > 1 then the funtion isincreasing.

5. The graph of f contains the points (1, 0), (a, 1),(

1a,−1

).

3

• ex. Letf(x) = 2− log3 (x + 1)

(a) Find the domain of f .

(b) Graph f .

(c) From the graph, determine the range and any asymptotes of f .

(d) Find f−1, the inverse of f .

4

(e) Use f−1 to find the range of f .

(f) Graph f−1.

• Solve:

a) log2(5x− 2) = 3

b) ln e−3x = 9

5

c) log5 25 = 8x + 4

d) e−3x+2 = 5

e) log4(x2 + x + 8) = 3

f) log2 8x = −3

6

Section 6.4: Example Answers

• ex. Letf(x) = 2− log3 (x + 1)

(b) Graph f .

(f) Graph f−1.

1

2

Section 6.5: Properties of Logarithms

• Properties of Logarithms:

1. loga 1 = 0

2. loga a = 1

3. aloga M = M

4. loga ar = r

5. loga(MN) = loga M + loga N

6. loga

(MN

)= loga M − loga N

7. loga M r = r loga M

8. If M = N , then loga M = loga N

9. If loga M = loga N , then M = N

10. Change of base formulas: loga M = logb Mlogb a

= log Mlog a

= ln Mln a

• ex. Find the exact value:

a) ln e√

5

b) log4 2 + log4 8

c) log5 7 · log7 25

1

• Write each expression as a sum and/or difference of logarithms. Expresspowers as factors.

a) ln x2

e2x

b) ln[

(x−4)2

x2−1

] 23, x > 4

• ex. Express as a single logarithm:

a) log(

x2+2x−8x2+x−6

)− log

(x2−2x−3

x+4

)

b) 24 log44√

x + log4(8x2)− log4 8

2

Section 6.6: Logarithmic and Exponential Equations

• ex. Solve:

a) 2 log3(x + 4)− log3 9 = 2

b) 22x + 2x+2 − 12 = 0

c) 3x = 14

d) 2x+1 = 51−2x

1

e) log2(3x + 2)− log4 x = 3

2