3.1 Exponents and Exponential Functions

3.1 Exponents and Exponential Functions

Exponential functions represent one of the most foundational elementary function families in the Gaokao Mathematics curriculum. They form the mathematical backbone for describing non-linear continuous growth, radioactive decay, compound interest, and complex physical dynamics. Mastery of exponential concepts requires a rigorous understanding of the transition from integer exponents to rational and real exponents, fundamental algebraic identities, structural properties of $y = a^x$, and analytical techniques for solving composite exponential equations and inequalities.

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1. Concept and Operational Rules of Exponents

Extension of Exponents from Integers to Real Numbers

Historically, exponentiation began as repeated multiplication for positive integers $a^n = \underbrace{a \cdot a \cdots a}_{n \text{ times}}$. To construct a continuous real function, the definition of exponents was systematically extended:

1. Zero and Negative Integer Exponents: For $a \neq 0$ and $n \in \mathbb{Z}^+$, $a^0 = 1$ and $a^{-n} = \frac{1}{a^n}$.
2. Rational Exponents: For $a > 0$, $m, n \in \mathbb{Z}^+$, and $n > 1$, the $n$-th root $\sqrt[n]{a}$ is extended to positive rational exponents as: $$a^{\frac{m}{n}} = \sqrt[n]{a^m}$$ Negative rational exponents are defined as: $$a^{-\frac{m}{n}} = \frac{1}{a^{\frac{m}{n}}} = \frac{1}{\sqrt[n]{a^m}}$$
3. Real Exponents: For an irrational number $\alpha$, $a^\alpha$ ($a > 0$) is defined as the unique limit of rational sequence approximations $a^{r_k}$ where rational $r_k \to \alpha$ as $k \to \infty$.

Prerequisite Base Restriction: When dealing with fractional or irrational exponents, the base $a$ MUST be strictly positive ($a > 0$). Expressions like $(-2)^{\frac{1}{2}} = \sqrt{-2}$ are undefined in real analysis $\mathbb{R}$.

Fundamental Algebraic Rules of Exponents

For any real bases $a > 0, b > 0$ and arbitrary real exponents $r, s \in \mathbb{R}$, the following operational identities hold unconditionally:

$$\begin{aligned} 1.\quad \& a^r \cdot a^s = a^{r+s} \\ 2.\quad \& \frac{a^r}{a^s} = a^{r-s} \\ 3.\quad \& (a^r)^s = a^{rs} \\ 4.\quad \& (ab)^r = a^r \cdot b^r \\ 5.\quad \& \left(\frac{a}{b}\right)^r = \frac{a^r}{b^r} = a^r b^{-r} \end{aligned}$$

High-Frequency Algebraic Transformations in Gaokao Problems

In algebraic simplifications involving exponential terms, quadratic symmetrical expansions are frequently tested:

$$\left(a^x + a^{-x}\right)^2 = a^{2x} + a^{-2x} + 2 \implies a^{2x} + a^{-2x} = \left(a^x + a^{-x}\right)^2 - 2$$ $$\left(a^x - a^{-x}\right)^2 = a^{2x} + a^{-2x} - 2 \implies \left(a^x + a^{-x}\right)^2 - \left(a^x - a^{-x}\right)^2 = 4$$

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2. Definition and Structural Properties of Exponential Functions

Mathematical Definition

A function of the form $f(x) = a^x$ (where $a > 0$ and $a \neq 1$) is defined as an exponential function with base $a$.

Why the restrictions $a > 0$ and $a \neq 1$?

• If $a \le 0$, $a^x$ is undefined for many real exponents (e.g., $(-4)^{1/2}$ is non-real).
• If $a = 1$, $f(x) = 1^x = 1$ degenerates into a constant linear function, lacking exponential growth dynamics.

Analytic Properties Summary Table

Graphical Symmetry and Base Comparison Rules

1. y-axis Symmetry: The graphs of $y = a^x$ and $y = \left(\frac{1}{a}\right)^x = a^{-x}$ are symmetric with respect to the $y$-axis.
2. Relative Base Ordering: In the first quadrant ($x > 0$), for exponential functions $f_1(x) = a^x, f_2(x) = b^x, f_3(x) = c^x, f_4(x) = d^x$, evaluating at $x = 1$ yields the direct height order of bases. The larger the base $a > 1$, the steeper the graph rises in Quadrant I.

3. Composite Exponential Functions $y = a^{f(x)}$ and Algebraic Substitution

Monotonicity of Composite Exponential Functions

For composite functions $y = a^{f(x)}$, monotonicity is determined by the composition of the outer exponential function $g(u) = a^u$ and the inner function $u = f(x)$ using the general composite monotonicity rule ("same direction yields increasing, opposite direction yields decreasing"):

• Case 1 ($a > 1$): $g(u) = a^u$ is strictly increasing. Thus, $y = a^{f(x)}$ has the exact same monotonicity intervals as $f(x)$.
• Case 2 ($0 < a < 1$): $g(u) = a^u$ is strictly decreasing. Thus, $y = a^{f(x)}$ has the reverse monotonicity intervals of $f(x)$.

Quadratic-Exponential Forms: $y = A \cdot a^{2x} + B \cdot a^x + C$

When dealing with expressions involving $a^{2x}$ and $a^x$, set $t = a^x$. Under this substitution:

1. The range restriction $t > 0$ MUST be explicitly imposed.
2. If $x \in [x_1, x_2]$, the interval for $t$ is bounded by $t \in [a^{x_1}, a^{x_2}]$ (if $a > 1$) or $t \in [a^{x_2}, a^{x_1}]$ (if $0 < a < 1$).
3. The problem reduces to finding the extremum of quadratic function $g(t) = A t^2 + B t + C$ over a bounded interval $t \in [t_1, t_2]$.

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4. Worked Gaokao Exam Examples

Example 1: Composite Monotonicity and Range Analysis

Problem: Find the strictly increasing interval and the range of the function $f(x) = \left(\frac{1}{2}\right)^{x^2 - 4x + 3}$.

Detailed Solution:

1. Define Inner Function: Let $u(x) = x^2 - 4x + 3 = (x-2)^2 - 1$.
   • Domain of $u(x)$ is $\mathbb{R}$.
   • Axis of symmetry for the quadratic function $u(x)$ is $x = 2$.
   • $u(x)$ is strictly decreasing on $(-\infty, 2]$ and strictly increasing on $[2, +\infty)$.
   • Minimum value of $u(x)$ is $u(2) = -1$, so range of $u(x)$ is $[-1, +\infty)$.
2. Analyze Outer Function: The outer function is $g(u) = \left(\frac{1}{2}\right)^u$.
   • Since base $a = \frac{1}{2} \in (0, 1)$, $g(u)$ is strictly decreasing on $\mathbb{R}$.
3. Determine Monotonicity of Composite Function:
   • On $(-\infty, 2]$, $u(x)$ decreases and $g(u)$ decreases, so $f(x) = g(u(x))$ is strictly increasing.
   • On $[2, +\infty)$, $u(x)$ increases and $g(u)$ decreases, so $f(x) = g(u(x))$ is strictly decreasing.
   • Therefore, the strictly increasing interval of $f(x)$ is $(-\infty, 2]$.
4. Determine Range:
   • Since $u \in [-1, +\infty)$ and $g(u) = (\frac{1}{2})^u$ is strictly decreasing:
   • Maximum value occurs at $u = -1$: $g(-1) = (\frac{1}{2})^{-1} = 2$.
   • As $u \to +\infty$, $g(u) \to 0^+$.
   • Thus, the range of $f(x)$ is $(0, 2]$.

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Example 2: Quadratic-Exponential Extremum with Restricted Domain

Problem: Given $x \in [-1, 2]$, find the maximum and minimum values of $f(x) = 4^x - 2^{x+1} + 2$.

Detailed Solution:

1. Algebraic Restructuring: Rewrite $f(x)$ using base $2^x$: $$f(x) = (2^x)^2 - 2 \cdot 2^x + 2$$
2. Variable Substitution: Set $t = 2^x$. Since $x \in [-1, 2]$ and $y = 2^x$ is strictly increasing: $$t \in \left[2^{-1}, 2^2\right] = \left[\frac{1}{2}, 4\right]$$
3. Formulate Quadratic Function in $t$: $$g(t) = t^2 - 2t + 2 = (t-1)^2 + 1, \quad t \in \left[\frac{1}{2}, 4\right]$$
4. Evaluate Extremum over Bounded Domain:
   • Vertex of parabola $g(t)$ is at $t = 1$, which lies inside $[\frac{1}{2}, 4]$.
   • At vertex $t = 1$ ($x = 0$): $g(1) = (1-1)^2 + 1 = 1$.
   • At left endpoint $t = \frac{1}{2}$ ($x = -1$): $g(\frac{1}{2}) = (\frac{1}{2}-1)^2 + 1 = \frac{1}{4} + 1 = \frac{5}{4} = 1.25$.
   • At right endpoint $t = 4$ ($x = 2$): $g(4) = (4-1)^2 + 1 = 9 + 1 = 10$.
5. Conclusion: The minimum value of $f(x)$ is $1$ (achieved at $x = 0$), and the maximum value is $10$ (achieved at $x = 2$).