Precalculus Study Guide: Functions, Exponentials, Logarithms, and Trigonometry

Test #2 Study Guide: Precalculus Topics

One-to-One Functions

Understanding one-to-one functions is essential for determining invertibility and analyzing function behavior.

• Definition: A function is one-to-one if each output value corresponds to exactly one input value.

• Testing One-to-One: Use the Horizontal Line Test on a graph; if any horizontal line intersects the graph more than once, the function is not one-to-one.

• Example: $f(x) = 2x + 3$ is one-to-one; $f(x) = x^2$ is not one-to-one over all real numbers.

Inverse Functions

Inverse functions reverse the effect of the original function, mapping outputs back to their inputs.

• Definition: The inverse of a function $f$ is a function $f^{-1}$ such that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

• Testing for Inverses: Two functions $f$ and $g$ are inverses if $f(g(x)) = x$ and $g(f(x)) = x$ for all $x$ in their domains.

• Finding Inverses: Solve $y = f(x)$ for $x$ in terms of $y$, then swap $x$ and $y$.

• Example: For $f(x) = 3x - 2$, the inverse is $f^{-1}(x) = \frac{x + 2}{3}$.

Exponential Functions

Exponential functions model rapid growth or decay and are fundamental in many applications.

• Definition: An exponential function has the form $f(x) = a \cdot b^x$, where $a \neq 0$, $b > 0$, $b \neq 1$.

• Graphing: Use transformations (translations, reflections, stretches) to graph variations of $f(x) = b^x$.

• Applications: Population growth, radioactive decay, compound interest.

• Example: $f(x) = 2^{x-1} + 3$ is a translation and stretch of $f(x) = 2^x$.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are used to solve for exponents.

• Definition: $y = \log_b(x)$ is the logarithm base $b$ of $x$, meaning $b^y = x$.

• Domain and Asymptotes: The domain is $x > 0$; the vertical asymptote is $x = 0$.

• Graphing: Use transformations to graph $\log_b(x)$ and its variations.

• Applications: pH in chemistry, Richter scale, sound intensity.

• Example: $f(x) = \log_2(x-1)$ is a translation of $\log_2(x)$.

Logarithms: Properties and Calculations

Logarithms have several properties that simplify expressions and calculations.

• Product Rule: $\log_b(xy) = \log_b(x) + \log_b(y)$

• Quotient Rule: $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$

• Power Rule: $\log_b(x^r) = r \log_b(x)$

• Change of Base Formula: $\log_b(x) = \frac{\log_k(x)}{\log_k(b)}$ for any base $k$.

• Calculator Use: Most calculators use base 10 ($\log$) or base $e$ ($\ln$).

Exponential and Logarithmic Equations

Solving equations involving exponentials and logarithms is crucial for many real-world problems.

• Exponential Equations: Set equal bases and solve for the exponent, or use logarithms to solve.

• Logarithmic Equations: Use properties of logarithms to combine terms and solve for the variable.

• Example: Solve $2^x = 8$ by rewriting as $2^x = 2^3$, so $x = 3$.

Applications: Exponential and Logarithmic Equations

Exponential and logarithmic equations are used to model growth and decay in various fields.

• Exponential Growth: $N(t) = N_0 e^{kt}$

• Exponential Decay: $N(t) = N_0 e^{-kt}$

• Example: Radioactive decay, population growth, compound interest.

Angles and Their Measure

Angles are measured in degrees and radians, and understanding their properties is foundational for trigonometry.

• Degree Measure: A full circle is $360^\circ$.

• Radian Measure: A full circle is $2\pi$ radians.

• Standard Position: Vertex at the origin, initial side along the positive $x$-axis.

• Coterminal Angles: Angles that share the same terminal side; differ by multiples of $360^\circ$ or $2\pi$.

• Example: $30^\circ$ and $390^\circ$ are coterminal.

Definitions of Trigonometric Functions

Trigonometric functions relate angles to ratios of sides in right triangles and points on the unit circle.

• Six Trigonometric Functions: Sine ($\sin$), Cosine ($\cos$), Tangent ($\tan$), Cosecant ($\csc$), Secant ($\sec$), Cotangent ($\cot$).

• Right Triangle Definitions:

  • $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

  • $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$

  • $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

• Special Angles: $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$

• Example: $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$

Graphs of Sine and Cosine Functions

Sine and cosine functions are periodic and can be transformed by changing amplitude, period, and phase shift.

• General Form: $y = A \sin(Bx + C) + D$

• Amplitude: $|A|$

• Period: $\frac{2\pi}{|B|}$

• Phase Shift: $-\frac{C}{B}$

• Transformations: Horizontal and vertical shifts, reflections, stretches.

Trigonometric Identities

Identities are equations true for all values in the domain and are used to simplify expressions and solve equations.

• Pythagorean Identities:

  • $\sin^2(x) + \cos^2(x) = 1$

  • $1 + \tan^2(x) = \sec^2(x)$

  • $1 + \cot^2(x) = \csc^2(x)$

• Reciprocal Identities:

  • $\csc(x) = \frac{1}{\sin(x)}$

  • $\sec(x) = \frac{1}{\cos(x)}$

  • $\cot(x) = \frac{1}{\tan(x)}$

• Quotient Identities:

  • $\tan(x) = \frac{\sin(x)}{\cos(x)}$

  • $\cot(x) = \frac{\cos(x)}{\sin(x)}$

Evaluating Trigonometric Functions

Trigonometric functions can be evaluated for special angles using reference triangles and the unit circle.

• Reference Angles: The acute angle formed with the $x$-axis.

• Special Values: $\sin(0) = 0$, $\sin(\frac{\pi}{2}) = 1$, $\cos(0) = 1$, $\cos(\frac{\pi}{2}) = 0$

• Example: $\tan(\frac{\pi}{4}) = 1$

Applications: Right Triangles

Trigonometry is used to solve for unknown sides and angles in right triangles.

• Angles of Elevation and Depression: Used in navigation, surveying, and physics.

• Solving Triangles: Use trigonometric ratios to find missing sides or angles.

• Example: If $\sin(\theta) = 0.6$, $\theta \approx 36.87^\circ$.

Sum and Difference Identities

Sum and difference identities allow calculation of trigonometric values for sums or differences of angles.

• Sine: $\sin(A \pm B) = \sin(A)\cos(B) \pm \cos(A)\sin(B)$

• Cosine: $\cos(A \pm B) = \cos(A)\cos(B) \mp \sin(A)\sin(B)$

• Tangent: $\tan(A \pm B) = \frac{\tan(A) \pm \tan(B)}{1 \mp \tan(A)\tan(B)}$

Trigonometric Equations

Solving trigonometric equations involves using identities and algebraic methods to find all solutions.

• Methods: Linear and quadratic techniques, using identities to simplify.

• Multiple-Angle and Half-Angle Equations: Use appropriate identities to solve.

• Example: Solve $2\sin(x) = 1$; $\sin(x) = \frac{1}{2}$, so $x = \frac{\pi}{6}, \frac{5\pi}{6}$ (plus coterminal angles).