IB Mathematical Analysis SL 1-2 (Check-in 2)

Domain and range of f(x) = 2^x?

Domain: (−∞,∞) Range: (0,∞)

Asymptote of f(x) = 2^x?

y = 0

Intercepts of f(x)=2^x?

(0,1)

Inverse of f(x)=2^x?

f^-1(x)=log_2(x)

Inverse of g(x)=e^x?

g^-1(x) = ln(x)

What is the general form of an exponential growth function?

f(x)=a⋅b^x, where a>0 b > 1

Exponential Growth

Occurs when b>1. The function increases rapidly.

Exponential Decay

Occurs when 0<b<1. The function decreases rapidly.

Logarithmic Function

The inverse of an exponential function, written y=log⁡_b(x)

Natural Logarithm (ln⁡)

A logarithm with base e (ln⁡(x)=log_⁡e(x)).

Compound Interest Formula

A = P(1 + r/n)^(nt)

Continuous Compound Interest Formula

A=P*e^(r*t)

Principal (P)

The original amount of money invested or borrowed.

Rate (r)

The annual interest rate expressed as a decimal.

Time (t)

The number of years the money is invested or borrowed.

Product Rule of Exponents

a^m * a^n = a^(m+n)

Quotient Rule of Exponents

a^m/a^n = a^(m-n)

Power Rule of Exponents

((a)^m)^n = a^mn

Product Rule of Logarithms

log⁡_b(MN)=log_⁡b(M)+log⁡_b(N)

Quotient Rule of Logarithms

log_b​(M/N​)=log_b(​M) − log_b(​N)

Horizontal Shift Rule

Vertical Shift Rule

f(x)+k shifts up k

Reflection Rule

Multiply by −1 to reflect across x-axis or y-axis.

Composite Function Definition

A function formed when one function is substituted into another, written as, (f*g)(x) = f(g(x))

Order of Composite Functions

(fg)(x) means apply g(x) first, then plug into f(x).

Inverse Function Definition

A function that “undoes” another function, satisfying f(f^-1(x))=x

How to Find an Inverse Algebraically

Replace f(x) with y, switch x and y, then solve for y

Graphical Relationship of Inverses

The graphs of a function and its inverse are reflections across the line y=x.

Domain and Range Rule for Inverses

The domain of f(x) becomes the range of f^−1(x), and vice versa.