From Middle School to High School: The Evolution of the Function Concept
In middle school, we focused on how one 'variable' changes in relation to another. However,Leibniz initially used 'function' to represent geometric quantities (coordinates, tangents, etc.) that change along a curve;Euler defined it as a relationship between variables; until Dirichlet proposed: If for every value of $x$, there is always a uniquely determined value of $y$ corresponding to it, then $y$ is a function of $x$. This shift marks the era of functions defined by 'correspondence'.
Think: Compare the middle school definition of functions with the set-based definition. What new insights do you have about functions?
In middle school, we focused on how one 'variable' changes in relation to another. However,Leibniz initially used 'function' to represent geometric quantities (coordinates, tangents, etc.) that change along a curve;Euler defined it as a relationship between variables; until Dirichlet proposed: If for every value of $x$, there is always a uniquely determined value of $y$ corresponding to it, then $y$ is a function of $x$. This shift marks the era of functions defined by 'correspondence'.
Think: Compare the middle school definition of functions with the set-based definition. What new insights do you have about functions?
Consistency Check for Functions: To determine if two functions are 'the same', both must satisfy:identical domains and identical correspondence rules. The choice of variable letters (e.g., $x$ or $t$) does not affect the essence of the function.
$$f: A \to B \text{ (Three Elements: Domain } A, \text{ Range } C \subseteq B, \text{ Correspondence Rule } f)$$
1. Gather polynomial terms: one $x^2$ square, three $x$ rectangular strips, and two $1\times1$ unit squares.
2. Begin assembling them geometrically.
3. They perfectly form a larger continuous rectangle! Width is $(x+2)$, height is $(x+1)$.
QUESTION 1
Find the domain of the function $f(x) = \frac{1}{4x+7}$.
$\{x \mid x \neq -\frac{7}{4}\}$
$\{x \mid x > -\frac{7}{4}\}$
$\{x \mid x \in \mathbb{R}\}$
$\{x \mid x \neq \frac{7}{4}\}$
Correct! According to the rule that the denominator of a fraction cannot be zero, $4x + 7 \neq 0 \Rightarrow x \neq -7/4$.
Incorrect. Remember the trap: when finding the domain, the denominator of a fraction cannot be zero.
QUESTION 2
Determine which pair of functions $f(x)$ and $g(x)$ are identical functions?
$f(x) = x - 1, g(x) = \frac{x^2}{x} - 1$
$f(x) = x^2, g(x) = (\sqrt{x})^4$
$f(x) = x^2, g(x) = \sqrt[3]{x^6}$
$f(x) = 1, g(x) = x^0$
Correct! For (3), $f(x) = x^2$ has domain $\mathbb{R}$, and $\sqrt[3]{x^6} = x^{6/3} = x^2$ also has domain $\mathbb{R}$. Other options have different domains.
Incorrect. The standard for determining 'identical functions' is that both domain and correspondence rule must be exactly the same.
QUESTION 3
Find the domain of the function $f(x) = \sqrt{1 - x} + \sqrt{x + 3} - 1$.
$[-3, 1]$
$(-3, 1)$
$(-\infty, 1]$
$[-3, +\infty)$
Correct! For even roots, the radicand must be non-negative: $1 - x \geq 0 \Rightarrow x \leq 1$ and $x + 3 \geq 0 \Rightarrow x \geq -3$. Taking the intersection gives $[-3, 1]$.
Incorrect. Note: for even roots, the radicand must be non-negative, and all root constraints must be satisfied simultaneously.
QUESTION 4
Are the functions $h = 130t - 5t^2$ and $y = 130x - 5x^2$ the same function?
Yes, the choice of variable letter does not affect the functional relationship
No, the independent variable letters differ
No, they have different physical meanings
Cannot determine, domain information is missing
Correct! The essence of a function lies in its correspondence rule and domain. Variable names ($t$ or $x$) are merely symbols and do not affect consistency.
Incorrect. Variable symbols are just carriers. As long as the domain and correspondence rule are consistent, they are the same function.
QUESTION 5
Find the domain of the function $f(x) = \frac{\sqrt{4 - x}}{x - 1}$.
$\{x \mid x \leq 4 \text{ and } x \neq 1\}$
$\{x \mid x < 4 \text{ and } x \neq 1\}$
$\{x \mid x \leq 4\}$
$\{x \mid x \neq 1\}$
Correct! The numerator requires $4 - x \geq 0 \Rightarrow x \leq 4$, and the denominator requires $x - 1 \neq 0 \Rightarrow x \neq 1$.
Incorrect. Both conditions—non-negative radicand and non-zero denominator—must be considered simultaneously.
QUESTION 6
In Example 3, which of the following functions is identical to $y = x$?
$y = (\sqrt{x})^2$
$u = \sqrt[3]{v^3}$
$y = \sqrt{x^2}$
$m = \frac{n^2}{n}$
Correct! $u = \sqrt[3]{v^3} = v$, with domain $\mathbb{R}$, matching $y = x$ exactly. (1) Domain is $[0, +\infty)$, (3) correspondence is $|x|$, (4) domain is $n \neq 0$.
Incorrect. Check each option’s domain. For example, $(\sqrt{x})^2$ requires $x \geq 0$.
QUESTION 7
The domain of the function $f(x) = \sqrt{x^5}$ is:
$[0, +\infty)$
$(0, +\infty)$
$\mathbb{R}$
$(-\infty, 0]$
Correct! $x^5 \geq 0 \Rightarrow x \geq 0$.
Incorrect. For even roots, $x^5$ must be greater than or equal to zero.
QUESTION 8
Find the domain of $f(x) = \frac{6}{x^2 - 3x + 2}$.
$\{x \mid x \neq 1 \text{ and } x \neq 2\}$
$\{x \mid x \neq 1 \text{ or } x \neq 2\}$
$\{x \mid x < 1 \text{ or } x > 2\}$
$\{x \mid 1 < x < 2\}$
Correct! The denominator $(x - 1)(x - 2) \neq 0$.
Incorrect. The denominator cannot be zero, so $x$ cannot equal any root of the equation.
QUESTION 9
The criterion for determining a function's graph is:
A vertical line perpendicular to the x-axis intersects the graph at most once
A vertical line perpendicular to the y-axis intersects the graph at most once
The graph must be a continuous curve
The graph must pass through the origin
Correct! According to the 'uniqueness' principle, each $x$ corresponds to only one specific $y$.
Incorrect. Consider: for every value of $x$, does $y$ always have a unique, well-defined value?
Challenge: Comprehensive Applications and Logical Reasoning with Functions
From Model Building to Rigorous Proof
Q1
A magazine is originally sold at 2.5 yuan per copy, with sales of 80,000 copies. Market research shows that for every 0.1 yuan increase in price, sales decrease by 2,000 copies. What price should be set to ensure total revenue after the price increase is no less than 200,000 yuan?
Solution Steps:
1. Let the price increase be $0.1x$ yuan ($x \geq 0$). Then the price is $2.5 + 0.1x$ yuan, and sales volume is $8 - 0.2x$ ten thousand copies.
2. Total revenue function: $y = (2.5 + 0.1x)(8 - 0.2x)$.
3. Set up the inequality: $(2.5 + 0.1x)(8 - 0.2x) \geq 20$.
4. Simplify: $20 - 0.5x + 0.8x - 0.02x^2 \geq 20 \Rightarrow 0.3x - 0.02x^2 \geq 0$.
5. Solve: $0 \leq x \leq 15$.
Conclusion: The price increase should be between 0 and 1.5 yuan, meaning the price should be between 2.5 and 4.0 yuan.
1. Let the price increase be $0.1x$ yuan ($x \geq 0$). Then the price is $2.5 + 0.1x$ yuan, and sales volume is $8 - 0.2x$ ten thousand copies.
2. Total revenue function: $y = (2.5 + 0.1x)(8 - 0.2x)$.
3. Set up the inequality: $(2.5 + 0.1x)(8 - 0.2x) \geq 20$.
4. Simplify: $20 - 0.5x + 0.8x - 0.02x^2 \geq 20 \Rightarrow 0.3x - 0.02x^2 \geq 0$.
5. Solve: $0 \leq x \leq 15$.
Conclusion: The price increase should be between 0 and 1.5 yuan, meaning the price should be between 2.5 and 4.0 yuan.
Q2
Tropical Storm Forecast: The storm center is located 600 km away from the port, in the southeast direction at $45^\circ$, moving northward at $20\text{ km/h}$. The impact radius is $450\text{ km}$. How long after will the port be affected? How long will the impact last?
Solution Steps:
1. Establish a coordinate system with the port at $(0,0)$. Initial position: $(300\sqrt{2}, -300\sqrt{2}) \approx (424.3, -424.3)$.
2. After $t$ hours, the coordinates are $(424.3, 20t - 424.3)$.
3. Distance squared: $d^2 = 424.3^2 + (20t - 424.3)^2 \leq 450^2$.
4. Solving: $(20t - 424.3)^2 \leq 22470 \Rightarrow |20t - 424.3| \leq 149.9$.
5. $13.7 \leq t \leq 28.7$.
Conclusion: Approximately $13.7$ hours later, the port will be affected, lasting about $15.0$ hours.
1. Establish a coordinate system with the port at $(0,0)$. Initial position: $(300\sqrt{2}, -300\sqrt{2}) \approx (424.3, -424.3)$.
2. After $t$ hours, the coordinates are $(424.3, 20t - 424.3)$.
3. Distance squared: $d^2 = 424.3^2 + (20t - 424.3)^2 \leq 450^2$.
4. Solving: $(20t - 424.3)^2 \leq 22470 \Rightarrow |20t - 424.3| \leq 149.9$.
5. $13.7 \leq t \leq 28.7$.
Conclusion: Approximately $13.7$ hours later, the port will be affected, lasting about $15.0$ hours.
Q3
Prove that the function $f(x) = -\frac{2}{x}$ is monotonically increasing on the interval $(-\infty, 0)$.
Proof Process:
1. Let $x_1, x_2 \in (-\infty, 0)$ such that $x_1 < x_2$.
2. Compute the difference: $f(x_1) - f(x_2) = -\frac{2}{x_1} - (-\frac{2}{x_2}) = \frac{2}{x_2} - \frac{2}{x_1} = \frac{2(x_1 - x_2)}{x_1x_2}$.
3. Sign determination: Since $x_1 < x_2$, $x_1 - x_2 < 0$; since $x_1, x_2 < 0$, $x_1x_2 > 0$.
4. Conclusion: $f(x_1) - f(x_2) < 0$, i.e., $f(x_1) < f(x_2)$. Therefore, the function is monotonically increasing on $(-\infty, 0)$.
1. Let $x_1, x_2 \in (-\infty, 0)$ such that $x_1 < x_2$.
2. Compute the difference: $f(x_1) - f(x_2) = -\frac{2}{x_1} - (-\frac{2}{x_2}) = \frac{2}{x_2} - \frac{2}{x_1} = \frac{2(x_1 - x_2)}{x_1x_2}$.
3. Sign determination: Since $x_1 < x_2$, $x_1 - x_2 < 0$; since $x_1, x_2 < 0$, $x_1x_2 > 0$.
4. Conclusion: $f(x_1) - f(x_2) < 0$, i.e., $f(x_1) < f(x_2)$. Therefore, the function is monotonically increasing on $(-\infty, 0)$.
Q4
A cylindrical log has a radius of $25\text{ cm}$. It is sawed into a rectangular timber. One side length is $x$, and area $y$ is expressed as a function of $x$.
Solution Steps:
1. The diagonal of the rectangle equals the cylinder diameter, $D = 50\text{ cm}$.
2. The other side of the rectangle is $\sqrt{50^2 - x^2}$.
3. Area: $y = x\sqrt{2500 - x^2}$.
4. Note the domain: $x \in (0, 50)$.
1. The diagonal of the rectangle equals the cylinder diameter, $D = 50\text{ cm}$.
2. The other side of the rectangle is $\sqrt{50^2 - x^2}$.
3. Area: $y = x\sqrt{2500 - x^2}$.
4. Note the domain: $x \in (0, 50)$.
✨ Key Points
For any $x$ in set $A$,uniquely corresponds to a $y$ in $B$.Focus on the core among the three elements,domainand correspondence.When checking identity, don't rush,domainsame domain is essential.
💡 Domain-First Principle
When finding the domain, the denominator of a fraction cannot be zero, and the radicand under an even root must be non-negative. Always clearly define the domain before analyzing function properties.
💡 Identifying Identical Functions
If the domain and correspondence rule are identical, they are the same function. Changing variable letters (e.g., $x$ to $t$) does not alter the function itself.
💡 Five-Step Method for Proving Monotonicity
Choose values ($x_1 < x_2$) → Compute difference ($f(x_1) - f(x_2)$) → Transform (factorization/common denominator) → Determine sign → Draw conclusion.
💡 Notes on Interval Notation
Solid dots correspond to closed intervals [ ], hollow dots to open intervals ( ). The infinity symbol $\infty$ always uses an open bracket.
💡 Modeling Real-World Problems
When solving real-world problems (e.g., personal income tax, displacement), always consider the physical meaning of variables, as this typically determines the function's domain.