college algebra :)

What is an exponential equation?

An equation where the variable is in the exponent, such as 2^(x+1)=8

What is the domain of a function with a denominator?

All real numbers except where the denominator equals 0.

What is the domain of a function with an even root (e.g., square root)?

All values of x where the expression under the root is ≥ 0.

What is the range of f(x) = e^x?

(0, ∞)

How do you find the vertex of a quadratic f(x) = ax^2 + bx + c?

Use x = -b / (2a) to find the x-coordinate. Plug back in to get the y.

What is the axis of symmetry of f(x) = ax^2 + bx + c?

x = -b / (2a)

How do you graph f(x) = a(x - h)^2 + k?

Vertex at (h, k), opens up if a > 0, down if a < 0. a controls the "width".

What is the general shape of an exponential function f(x) = a * b^x?

If a > 0 and b > 1, it increases. If 0 < b < 1, it decays.

How do you graph a rational function with a vertical asymptote?

Set the denominator equal to 0 and solve.

How do you find holes in a rational function?

Factor numerator and denominator. If a factor cancels, set it equal to 0 — that's the x-value of the hole.

What’s the end behavior of a rational function f(x) = p(x)/q(x)?

• If deg(p) < deg(q): HA at y = 0

• If deg(p) = deg(q): HA at ratio of leading coefficients

• If deg(p) > deg(q): no HA; possible slant asymptote

What is the Quadratic Formula?

x = (-b ± sqrt(b^2 - 4ac)) / (2a)

What does the discriminant D = b^2 - 4ac tell you?

• D > 0: 2 real solutions

• D = 0: 1 real solution

• D < 0: 2 complex solutions

What’s a common method to factor ax^2 + bx + c?

Use the “ac method”:

1. Find 2 numbers that multiply to a * c and add to b.

2. Split the middle term and factor by grouping.

What is factoring by grouping?

Group terms into pairs and factor each group, then factor out the common binomial.

When should you complete the square?

Use when solving quadratics or converting to vertex form:
f(x) = a(x - h)^2 + k

How do you find the y-intercept of any function?

Plug in x = 0

How do you find the x-intercepts (zeros) of a function?

Set f(x) = 0 and solve for x.

What is the domain of f(x) = log_b(x)?

What is the inverse of f(x) = a * log_b(x - h) + k?

f^(-1)(x) = b^((x - k)/a) + h

What transformation does f(x + 3) cause?

Shift left 3 units.

What transformation does -f(x) cause?

Reflect across the x-axis.

What is the vertical asymptote of f(x) = log_b(x - c)?

x = c

What is the horizontal asymptote of f(x) = a * e^(bx) + c?

y = c

How do you simplify f(g(x))?

Replace every x in f(x) with g(x)

What does (f ∘ g)(x) mean?

Apply g first, then apply f to that result: f(g(x))

What are common domain mistakes when simplifying?

Canceling factors and forgetting to check their original restrictions (e.g., don't ignore excluded x-values even if they cancel out later).

What is the general shape of f(x) = log_b(x)?

Increases slowly, passes through (1, 0), and has a vertical asymptote at x = 0.

What does log_b(a) = c mean in exponential form?

b^c = a

What is the one-to-one property of exponents?

If b^m=b^n, then m=n, provided b>0, b≠1.

What is a logarithmic equation?

An equation that contains a logarithm with a variable in its argument, such as log⁡2(x+3)=4

What is the domain of a logarithmic function?

The set of values where the argument of the log is positive.

Steps to solve b^m =b^n

Use the one-to-one property to equate exponents: m=n

What do you do if bases are not the same in b^m=a

Take logarithms of both sides and apply log properties.

How do you solve an equation like 2^x=10

Take the log of both sides: xlog⁡2=log⁡10 then solve for x

What is the general strategy to solve a logarithmic equation?

• Combine logs.

• Convert to exponential form.

• Solve.

• Check domain restrictions.

Why must you check solutions to logarithmic equations?

To ensure no undefined values (i.e., log of non-positive numbers).

What is the basic shape of f(x)=b^x

Curves upward, approaching 0 as X→ -infinity, increasing as x → infinity

What is the horizontal asymptote of f(x)=b^x+c

y=c

How do you find the y-intercept of an exponential function?

Plug in x=0

What is the basic shape of f(x)=log⁡_b(x)

Increases slowly, with vertical asymptote at x=0

How do you transform f(x)=log_⁡b(x−h)+k

Shift right by hh, up by kk; vertical asymptote is x=h

Domain of f(x)=log⁡(x−2)

x>2

What is the general formula for exponential growth?

P(t)=P_0​^ert, where P_0 is the initial amount, r is the growth rate, and t is time.

What is the general formula for exponential decay?

V(t)=V_0​e−rt, where V0V0​ is the initial amount and rr is the decay rate.

What does f(2x) mean in function notation?

Multiply the input x by 2 before applying the function.

What does 2f(x) mean?

Multiply the entire function output f(x) by 2.

Difference between f(2x), f(x) + 2, and f(x) + f(2)?

• f(2x): Plug in 2x into the function.

• f(x) + 2: Add 2 to the function output.

• f(x) + f(2): Add f(x) and f(2) together.

Define the sum of two functions f and g.

(f + g)(x) = f(x) + g(x)

Define the difference of two functions f and g.

(f - g)(x) = f(x) - g(x)

Define the product of two functions f and g.

(f * g)(x) = f(x) * g(x)

Define the quotient of two functions f and g.

(f / g)(x) = f(x) / g(x), as long as g(x) ≠ 0

What is the formula for the difference quotient of a function f?

[f(x + h) - f(x)] / h

What is the composition (g ∘ f)(x)?

g(f(x)) — apply f first, then g to the result.

How do you find the domain of a composite function g(f(x))?

Include all x such that x is in the domain of f and f(x) is in the domain of g.

What does it mean if a function is one-to-one?

Each output corresponds to only one input.

What test determines if a function is one-to-one?

The Horizontal Line Test.

How do you verify two functions are inverses of each other?

• (g ∘ f)(x) = x

• (f ∘ g)(x) = x

What is the graphical relationship between a function and its inverse?

They are reflections over the line y = x.

How do you find the inverse of a function f(x)?

• Replace f(x) with y

• Switch x and y

• Solve for y

• Rename y as f^(-1)(x)