Calculus I · Chapter 2
Chapter 2: Nonlinear Functions
Domain & range · Quadratics · Polynomials & rational functions · Exponentials · Logarithms · Growth, decay & finance
Linear functions describe constant change. The world, however, is full of quantities that accelerate, level off, oscillate, or explode. This chapter assembles the complete function library you'll differentiate and integrate for the rest of the course: quadratics, polynomials, rational functions, and, most importantly for applications, the exponential and logarithmic functions that govern growth, decay, and money.
Section 2.1
Properties of Functions
Functions, Domain, and Range
Definition: Function
A function $f$ is a rule that assigns to each input $x$ exactly one output $f(x)$. The domain is the set of all allowable inputs; the range is the set of all resulting outputs.
Finding the Domain: Two Rules
Assume the domain is all real numbers except values that:
- make a denominator zero (division by zero is undefined), or
- put a negative under an even root (e.g. $\sqrt{\,\cdot\,}$ requires its inside $\geq 0$).
The Vertical Line Test
A graph represents a function if and only if no vertical line crosses it more than once. (More than one crossing would mean one input with two outputs, not allowed.)
📘 Example 2.1: Domain of a Function
Find the domain of $f(x) = \dfrac{\sqrt{x+4}}{x-2}$.
- Even root requires $x + 4 \geq 0 \Rightarrow x \geq -4$.
- Denominator requires $x - 2 \neq 0 \Rightarrow x \neq 2$.
- Combine: $x \geq -4$ and $x \neq 2$.
Even and Odd Functions
Definition: Symmetry
A function is even if $f(-x) = f(x)$ (graph is symmetric about the $y$-axis, like $x^2$). It is odd if $f(-x) = -f(x)$ (symmetric about the origin, like $x^3$). Most functions are neither.
Section 2.2
Quadratic Functions: Translation & Reflection
A quadratic function $f(x) = ax^2 + bx + c$ (with $a \neq 0$) graphs as a parabola. The vertex form reveals everything about its shape and position at a glance.
Vertex Form
$$f(x) = a(x - h)^2 + k$$
The vertex is $(h, k)$. The parabola opens up if $a > 0$ (vertex is a minimum) and down if $a < 0$ (vertex is a maximum). Larger $|a|$ makes it narrower. The axis of symmetry is the vertical line $x = h$. From standard form, the vertex sits at $x = -\dfrac{b}{2a}$.
The parabola $y = a(x-h)^2 + k$. Drag the sliders to translate ($h$, $k$) and reflect/stretch ($a$). The vertex follows $(h,k)$; flipping the sign of $a$ reflects it across the horizontal axis.
📘 Example 2.2: Completing the Square to Find the Vertex
Write $f(x) = 2x^2 - 8x + 5$ in vertex form and find its minimum.
- Factor $2$ from the $x$-terms: $f(x) = 2(x^2 - 4x) + 5$.
- Complete the square inside: $x^2 - 4x = (x-2)^2 - 4$.
- Substitute: $f(x) = 2[(x-2)^2 - 4] + 5 = 2(x-2)^2 - 3$.
Section 2.3
Polynomial & Rational Functions
Definition: Polynomial Function
$$f(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0, \quad a_n \neq 0.$$
$n$ is the degree and $a_n$ the leading coefficient. The degree caps the number of turning points at $n - 1$ and the number of real roots at $n$.
End Behavior: Determined by the Leading Term
For large $|x|$, a polynomial behaves like its leading term $a_n x^n$:
- Even degree: both ends point the same way, up if $a_n > 0$, down if $a_n < 0$.
- Odd degree: ends point opposite ways, up-right if $a_n > 0$, down-right if $a_n < 0$.
Definition: Rational Function
A ratio of two polynomials, $f(x) = \dfrac{p(x)}{q(x)}$ with $q(x) \neq 0$. Key features:
- Vertical asymptotes where $q(x) = 0$ but $p(x) \neq 0$.
- Horizontal asymptote determined by comparing the degrees of $p$ and $q$.
| Degree comparison | Horizontal asymptote |
|---|---|
| $\deg p < \deg q$ | $y = 0$ |
| $\deg p = \deg q$ | $y = \dfrac{\text{leading coeff of }p}{\text{leading coeff of }q}$ |
| $\deg p > \deg q$ | none (grows without bound) |
📘 Example 2.3: Asymptotes of a Rational Function
Find all asymptotes of $f(x) = \dfrac{3x + 2}{x - 4}$.
- Vertical: denominator zero at $x = 4$ (numerator $\neq 0$ there) → $x = 4$.
- Horizontal: degrees equal, ratio of leading coefficients $= \tfrac{3}{1}$ → $y = 3$.
Sponsored
Section 2.4
Exponential Functions
Definition: Exponential Function
$$f(x) = a^x, \quad a > 0,\; a \neq 1.$$
The variable is in the exponent. If $a > 1$ the function grows; if $0 < a < 1$ it decays. Every such graph passes through $(0, 1)$, stays positive, and has the $x$-axis as a horizontal asymptote.
The Natural Base $e$
The special constant $e \approx 2.71828$ arises as the limit of $\left(1 + \tfrac{1}{n}\right)^n$ as $n \to \infty$. The function $e^x$ is the cornerstone of calculus because, as we'll see in Chapter 4, it is its own derivative.
$y = a^x$. Slide the base $a$ across $1$ to watch growth ($a>1$) flip into decay ($a<1$). All curves pass through $(0,1)$ and hug the $x$-axis on one side.
📘 Example 2.4: Solving an Exponential Equation
Solve $9^{x} = 27^{\,x-1}$.
Rewrite both sides with the same base
$9 = 3^2$ and $27 = 3^3$, so $(3^2)^x = (3^3)^{x-1} \Rightarrow 3^{2x} = 3^{3x - 3}$.
Equal bases ⇒ equal exponents
$2x = 3x - 3 \Rightarrow x = 3$.
$x = 3$
Section 2.5
Logarithmic Functions
Definition: Logarithm
The logarithm is the inverse of the exponential:
$$y = \log_a x \iff a^y = x.$$
It answers "what power of $a$ gives $x$?" The natural log $\ln x = \log_e x$ is the inverse of $e^x$. Domain: $x > 0$. The graph is the reflection of $a^x$ across the line $y = x$.
$e^x$ (blue) and its mirror image $\ln x$ (green) reflected across the dashed line $y = x$. Inverses undo each other: the point $(0,1)$ on $e^x$ becomes $(1,0)$ on $\ln x$.
Properties of Logarithms
For $x, y > 0$:
- Product: $\log_a(xy) = \log_a x + \log_a y$
- Quotient: $\log_a\!\dfrac{x}{y} = \log_a x - \log_a y$
- Power: $\log_a(x^r) = r\log_a x$
- Change of base: $\log_a x = \dfrac{\ln x}{\ln a}$
📘 Example 2.5: Solving with Logarithms
Solve $5e^{0.2t} = 40$ for $t$.
- Isolate the exponential: $e^{0.2t} = 8$.
- Take $\ln$ of both sides: $0.2t = \ln 8$.
- Solve: $t = \dfrac{\ln 8}{0.2} \approx \dfrac{2.079}{0.2} \approx 10.4$.
Section 2.6
Applications: Growth, Decay & Mathematics of Finance
Exponential Growth & Decay Model
$$y = y_0 e^{kt}$$
$y_0$ is the initial amount, $t$ is time, and $k$ is the continuous rate. Growth when $k > 0$; decay when $k < 0$. Used for populations, radioactive decay, drug elimination, and continuous interest.
Compound Interest: Two Formulas
| Compounding | Formula | Symbols |
|---|---|---|
| $m$ times per year | $A = P\left(1 + \dfrac{r}{m}\right)^{mt}$ | $P$ principal, $r$ annual rate, $t$ years |
| Continuous | $A = Pe^{rt}$ | the $m \to \infty$ limit |
📘 Example 2.6: Continuous Compound Interest
\$5000 is invested at $6\%$ compounded continuously. What is it worth after $10$ years?
Apply $A = Pe^{rt}$
$$A = 5000\,e^{(0.06)(10)} = 5000\,e^{0.6} \approx 5000(1.8221) \approx 9110.59.$$
$A \approx \$9110.59$
📘 Example 2.7: Radioactive Decay & Half-Life
A substance decays with $k = -0.023$ per year. What fraction remains after $30$ years, and what is its half-life?
Fraction remaining
$\dfrac{y}{y_0} = e^{-0.023(30)} = e^{-0.69} \approx 0.502.$ About half remains.
Half-life: solve $e^{-0.023t} = 0.5$
$-0.023t = \ln 0.5 \Rightarrow t = \dfrac{-0.693}{-0.023} \approx 30.1$ years.
≈ $50.2\%$ remains; half-life ≈ $30.1$ years
Sponsored
Looking Ahead
We now have the full cast of functions. In Chapter 3 we make precise what it means for these curves to have a slope at a single point, the limit, and define the derivative. The exponential and log functions you just met will reappear as the cleanest derivatives in all of calculus.