MATH 1500: Essentials - Inverses, Transformations, & Exponential/Log Functions

What does it x̄ for a f(x) to be invertible?

1. A f(x) f = invertible if there exists a f(x) g such that (f ∘ g)(x) = x = (g ∘ f)(x).

2. If such a g exists, it = unique, + = denoted as f^-1.

What does f^-1 “undo” x̄, in terms of f(a) = b?

1. (f ∘ f^-1)(x) = x = (f^-1 ∘ f)(x) by def.

2. This = equivalent to saying f(a) = b iff f^-1(b) = a.

3. So f^-1 “undoes” f, + vice versa.

How does the graph of f^-1 relate to the graph of f?

1. f(a) = b iff f^-1(b) = a.

2. So the pt (a,b) lies on the graph of f iff (b,a) lies on the pt on the graph of f^-1.

Example: if (-1, 5) lies on the graph of f, then (5, -1) lies on the graph of f^-1.

How do you find the inverse of a function?

1. Define f(x).

2. Swap “x” and “y.” → find y → y = f^-1(x).

How do you determine if functions, e.g. f(x) and g(x) are the inverse of each other?

1. Solve for f(g(x)).

2. Solve for g(f(x)).

3. If f(g(x)) = x and if g(f(x)) = x, then they are inverses of each other.

How do you determine if f^-1(x) is a function?

If f(x) passes the horizontal line test (i.e. horizontal line passes through the function only once), then f^-1(x) is a function.

How to solve a repeating chain like f(f⁻¹(f(f⁻¹(f(f⁻¹(1))))))?

label the innermost f⁻¹(1) as unknown (call it A)

↓ f(A) cancels to 1

↓ next layer out is f⁻¹(1) again → still A

↓ f(A) cancels to 1 again

↓ repeat for every f/f⁻¹ pair in the chain

∴ answer stays 1 the whole way through, never need to know A

How to find the slope between two points?

m = (y₂ − y₁)/(x₂ − x₁)

General equation of a line, once you know the slope?

y = mx + b

↑ b is the y-intercept, still unknown until solved

How to find b once you have m and one point (x₁, y₁)?

plug the point into y = mx + b

↓ solve for b

What's the relationship between slopes of perpendicular lines?

m₁ · m₂ = −1

∴ m₂ = −1/m₁ (negative reciprocal — flip it and flip the sign)

How to find the equation of a line perpendicular to another, through a given point?

1. Find the slope of the given line (using the two points)

2. Take the negative reciprocal → that's your new slope

3. Plug the new slope + given point into y = mx + b, solve for b

4. Write y = mx + b

How do you determine if a line is above a specific point (x₀, y₀)?

1. Find the equation of the line in the form y = f(x).

2. Plug the point’s x-coord (x₀) → the line equation to find the line’s hight @ the spot: y_line = f(x₀).

3. Compare the results: if y_line > y₀, the line = above the pt.

How do you determine the shift when g(x) = f(x) + a? (vertical shifts!)

1. If (x, y) is a point on f(x), then (x, y + a) is the corresponding point on g(x), since g(x) = f(x) + a = y + a.

2. If a is positive, the graph shifts up |a| units.

3. If a is negative, the graph shifts down |a| units.

How do you determine the shift when g(x) = f(x − c)? (horizontal shifts!)

1. If (x, y) is a point on f(x), then (x + c, y) is the corresponding point on g(x), since g(x + c) = f((x + c) − c) = f(x) = y.

2. If c is positive, the graph shifts right |c| units.

3. If c is negative, the graph shifts left |c| units.

What = an exponential f(x)?

f(x) = a^x, only if a > 0

The # a = based of the exponential f(x).

What = the logarithmic f(x), + how does it relate to exponentials?

1. Let f(x) = a^x for a > 0. Since f(x) = injective, it has an inverse.

2. The inverse of f(x) = dented as log_ax, called the logarithmic f(x) w/ base a.

Example: f(x) = 2^x → f^-1(x) = log₂x

What identities come from f and f⁻¹ being inverses, when f(x) = aˣ and f⁻¹(x) = logₐ(x)?

1. Since f(f⁻¹(x)) = x = f⁻¹(f(x)), substitute in aˣ and logₐ(x).

2. This gives logₐ(aˣ) = x and aˡᵒᵍᵃ⁽ˣ⁾ = x.

How do you convert between logarithmic and exponential form?

• y = logₐx is equivalent to aʸ = x.

Example: log₃z = r is equivalent to 3ʳ = z.

What is log_a1, for nay base a > 0, a ≠1?

• Let y = log_a1 → a^y = 1.

• Since 1 = a⁰, we have a^y = a⁰.

• Equating exponents gives y = 0.

• So log_a1 = 0, for any valid base a.

What’s the common logarithm (base 10) notation?

log₁₀(x) = log(x)

What’s the natural logarithm, and how does it relate to e^x?

• The natural exponential f(x) = e^x.

• Its inverse = the natural logarithm, denoted ln(x).

• log_e(x) = ln(x).

What’s the key rule for logs and their graphs?

• a > 1 → log_ax ↑ (goes up as x ↑)

• 0 < a < 1 → log_ax ↓ (goes down as x ↑)

Product rule for logs?

• log_a(mn) = log_a(m) + log_a(n)

Multiplying inside the log becomes adding outside.

Quotient rule for logs?

• log_a(m/n) = log_a(m) - log_a(n).

Dividing inside the log becomes subtracting outside.

Exponent rule for logs?

• log_a(m^k) = k × log_a(m)

An exponent inside log becomes a multiplier outside.

Change of base formula?

• log_a(x) = log (x)/log(a) = ln(x)/ln(a)

Used to compute a log in any base using a calculator (which only has log and ln).