Standard Functions and Their Graphs

Key Functions and Their Properties

This section introduces some of the most important real functions used in mathematics.

• Identity Function:

  • Definition: $f(x) = x$.
  • Domain: $\mathbb{R}$, Range: $\mathbb{R}$

• Constant Function:

  • Definition: $f(x) = c$, where c is a constant.
  • Domain: $\mathbb{R}$, Range: $\{c\}$

• Modulus Function:

  • Definition: $f(x) = |x| = \begin{cases}x, & x \ge 0\\ -x, & x < 0\end{cases}$
  • Domain: $\mathbb{R}$, Range: $[0, \infty)$

• Signum Function:

  • Definition: $f(x) = \begin{cases}1, & if\ x > 0\\ 0, & if\ x = 0\\ -1, & if\ x < 0\end{cases}$
  • Domain: $\mathbb{R}$, Range: $\{-1, 0, 1\}$

• Greatest Integer Function (GIF):

  • Definition: $f(x) = [x]$. It returns the greatest integer less than or equal to x.
  • Domain: $\mathbb{R}$, Range: $\mathbb{Z}$ (the set of all integers).

• Even and Odd Functions:

  • Even Function: A function is even if $f(-x) = f(x)$ for all x in its domain (e.g., $f(x)=x^2$). Its graph is symmetric about the y-axis.
  • Odd Function: A function is odd if $f(-x) = -f(x)$ for all x in its domain (e.g., $f(x)=x^3$). Its graph is symmetric about the origin.

Example 1: Domain of a Combined Function

Question: Find the domain of the function $f(x) = \sqrt{x-2} + \frac{1}{\log_{10}(4-x)}$.

Solution: We need to satisfy three conditions:

1. For $\sqrt{x-2}$ to be defined, $x-2 \ge 0 \implies x \ge 2$.
2. For $\log_{10}(4-x)$ to be defined, $4-x > 0 \implies x < 4$.
3. For the denominator not to be zero, $\log_{10}(4-x) \ne 0 \implies 4-x \ne 1 \implies x \ne 3$. Combining all conditions, we need $x \ge 2$, $x < 4$, and $x \ne 3$. The domain is $[2, 4) - \{3\}$, which can also be written as $[2, 3) \cup (3, 4)$.

Example 2: Range of a Modulus Function

Question: Find the range of $f(x) = |x-1| + |x-3|$.

Solution: We analyze this piecewise. The critical points are x=1 and x=3.

• Case 1: $x < 1$: $f(x) = -(x-1) - (x-3) = -2x+4$. As $x \to 1^-$, $f(x) \to 2$. As $x \to -\infty$, $f(x) \to \infty$. So on this interval, the range is $(2, \infty)$.
• Case 2: $1 \le x \le 3$: $f(x) = (x-1) - (x-3) = 2$. On this interval, the function is constant at 2.
• Case 3: $x > 3$: $f(x) = (x-1) + (x-3) = 2x-4$. As $x \to 3^+$, $f(x) \to 2$. As $x \to \infty$, $f(x) \to \infty$. So on this interval, the range is $(2, \infty)$. Combining all cases, the minimum value is 2, and it can go to infinity. The range is $[2, \infty)$.

Example 3: Solving an Equation with GIF

Question: Solve the equation $[x]^2 - 5[x] + 6 = 0$, where $[x]$ is the Greatest Integer Function.

Solution: Let $y=[x]$. The equation becomes $y^2-5y+6=0$. Factoring gives $(y-2)(y-3)=0$. So, $y=2$ or $y=3$. This means $[x]=2$ or $[x]=3$.

• If $[x]=2$, then by definition, $2 \le x < 3$.
• If $[x]=3$, then by definition, $3 \le x < 4$. The complete solution set is the union of these two intervals: $[2, 4)$.

Example 4: Even and Odd Functions

Question: Determine if the function $f(x) = \log(\frac{1-x}{1+x})$ is even, odd, or neither.

Solution: We evaluate $f(-x)$: $$f(-x) = \log\left(\frac{1-(-x)}{1+(-x)}\right) = \log\left(\frac{1+x}{1-x}\right)$$ Using the logarithm property $\log(a/b) = -\log(b/a)$: $$f(-x) = -\log\left(\frac{1-x}{1+x}\right) = -f(x)$$ Since $f(-x) = -f(x)$, the function is odd.

Example 5: Range of a Rational Function

Question: Find the range of the function $f(x) = \frac{x^2+x+1}{x^2-x+1}$.

Solution: Let $y = \frac{x^2+x+1}{x^2-x+1}$. Cross-multiply: $y(x^2-x+1) = x^2+x+1 \implies (y-1)x^2 - (y+1)x + (y-1) = 0$. Since x is a real number, the discriminant of this quadratic in x must be greater than or equal to zero. $D = (-(y+1))^2 - 4(y-1)(y-1) \ge 0$. $(y+1)^2 - 4(y-1)^2 \ge 0$. $(y^2+2y+1) - 4(y^2-2y+1) \ge 0$. $-3y^2+10y-3 \ge 0 \implies 3y^2-10y+3 \le 0$. Factoring gives $(3y-1)(y-3) \le 0$. The roots are $y=1/3$ and $y=3$. Since the parabola is upward-opening, the expression is less than or equal to 0 between the roots. The range is $[1/3, 3]$.