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MTH111Z Precalculus 1 - Functions

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Educational resources and practice exercises for MTH111Z Precalculus 1, focusing on functions. Topics include power, exponential, and logarithmic functions, polynomial behavior, the Fundamental Theorem of Algebra, function transformations, composition, and mathematical modeling using scatterplots and data patterns.

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What's inside MTH111Z Precalculus 1

  1. Understand Exponential Growth and Decay

    Exponential growth occurs when each new value in a sequence is a multiple of the previous one, resulting in a rapid increase (e.g., doubling, tripling). Examples include viral spread, nuclear chain reactions, and cell division. Exponential decay occurs when each new value is a fraction of the previous one (e.g., halving), resulting in a rapid decrease, such as the elimination of teams in a tournament bracket.

  2. Use logarithmic scales for measurement

    A logarithmic scale is created by using logarithms as increments on a number line. Unlike linear scales, values on a logarithmic scale are spread out for small numbers and compressed for larger numbers, allowing for the visualization of data across vast ranges of magnitude.

  3. Apply the concept of Direct Variation

    Two quantities $x$ and $y$ vary directly (or are directly proportional) if $y$ is a constant multiple of $x$, expressed as $y = kx$, where $k$ is the constant of variation. This concept extends to power functions where $y$ varies directly with the $p$-th power of $x$ ($y = kx^p$). For example, the volume of a sphere $V=\frac{4}{3} \pi r^{3}$ means volume is proportional to the cube of the radius with a constant of variation $k = \frac{4}{3} \pi$.

  4. Determine the total number of zeros for a polynomial

    According to the Fundamental Theorem of Algebra, a polynomial of degree $n$ will always have exactly $n$ zeros. These zeros may include real numbers, imaginary numbers, or repeated roots.
    Example: $p(x) = 2x^4 - 3x + 8$ is a 4th degree polynomial, therefore it has exactly 4 zeros.

  5. Understand the mathematical modeling process

    A mathematical model is a simplified representation of a real-world object, used when working with the actual object is impractical, unsafe, or too expensive. The process involves:

    1. Gathering a set of data points and a description of the scenario.
    2. Finding a function equation that closely aligns with observations.
    3. Using mathematical rules to analyze behavior and make predictions.
    4. Refining the model through continued observation to increase accuracy.

  6. Understand the mathematical definition and properties of the constant e

    The number $e$ is an irrational constant approximately equal to $2.718$. It is defined as the limit of the expression $(1+h)^{1/h}$ as $h$ approaches $0$. This constant is fundamental to science and mathematics, serving as the base for 'the' exponential function $f(x) = e^x$.
    (1+h)^{1/h} \rightarrow e \text{ as } h \rightarrow 0

  7. Understand the concept of function composition

    Function composition occurs when the output of one function is used as the input to another function. If you have two functions, $f$ and $g$, the composition involves taking a value $x$, applying function $g$ to get a result, and then applying function $f$ to that result.

  8. Graph and analyze composite functions

    Analyzing the graph of a composite function $h = f \circ g$ by looking at the individual graphs of $f$ and $g$ is complex. The recommended approach is to first derive the algebraic equation for $h(x)$ and then use a graphing utility to analyze its behavior, such as identifying maximums, asymptotes, and intervals of increase/decrease.
    Example: f(x) = e^{-x}, g(x) = x^2
Composite: h(x) = e^{-(x^2)}
Analysis via graph:
- Domain: All real numbers
- Range: (0, 1]
- Horizontal Asymptote: x-axis (y=0)
- Max: h(0) = 1

  9. Use scatterplots for data visualization

    A scatterplot is a tool used to visualize patterns in data by plotting pairs of data as points on a grid. It allows researchers to identify trends (e.g., the time of day tornadoes most frequently form) and create models (curves or lines) that can predict values for points where no data was explicitly recorded.

  10. Best practices for students asking for tutoring help

    Students seeking help from human tutors or AI assistants should follow these steps to get the most effective support:

    1. State the goal: Clearly identify the section and the specific task (e.g., "I’m working in Section 4.3 and need to find the composition of two functions.").
    2. Share work: Provide the problem and a description or photo of the steps already attempted.
    3. Be specific: Pinpoint exactly where the confusion lies (e.g., "I'm stuck on the part where I substitute $g(x)$ into $f(x)$.").
    4. Ask for coaching: Request help understanding the concept rather than just the answer to enable independent problem solving.

  11. Determine function concavity

    Concavity describes the curvature of a graph. A graph is concave up if it bends upward (like a cup) and concave down if it bends downward (like a frown). Straight lines have no concavity. Concavity indicates the rate of change: an increasing, concave down function has a slowing growth rate, while an increasing, concave up function has an accelerating growth rate.

  12. Evaluate common and natural logarithms

    Logarithms can be evaluated by rewriting them as exponential equations or using a calculator. Two standard logarithms are built into most scientific calculators:

    • Common Logarithm: Base 10, written as $\log(x)$.
    • Natural Logarithm: Base $e$, written as $\ln(x)$.
    Example 1: log 1000 = 3 (because 10^3 = 1000)
Example 2: ln 1000 ≈ 6.90776 (because e^6.90776 ≈ 1000)