Function

The set of all possible values of the independent variable (\(x\)) in a function is called the domain of the function.

You should always give the domain in form of set or interval.

• The domain of a polynomial function \(f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0\) is the set of all real numbers \(\mathbb{R}\).

  The domain of \(f(x) = 5x^4 + x^3 - 3x^2 + 11\) is \(\mathbb{R}\).

• The domain of a fraction function \(f(x) = 1/g(x)\) is the set of all points such that the denominator is non-zero, ie, \(\{x : g(x) \neq 0\}\).

  The domain of \(f(x) = \frac{1}{x-3}\) is \(\mathbb{R}\setminus\{3\}\).

• The domain of a square root function \(f(x) = \sqrt{g(x)}\) is the set of all points such that the term under the square root symbol non-negative, ie, \(\{x : g(x) \geq 0\}\).

  The domain of \(f(x) = \sqrt{x^2 - 9}\) is \((-\infty, -3] \cup [3,\infty)\)

• The domain of a logarithmic function \(f(x) = log_a g(x), a > 0, a\neq 1\) is the set of all points such that \(g(x) > 0\), ie, \(\{x : g(x) > 0\}\)

  The domain of \(f(x) = ln (x^2 - 9)\) is \((-\infty, -3) \cup (3,\infty)\)

• The domain of an exponential function \(f(x) = a^x, a > 0\) and \(a\neq 1\) is the set of all real numbers \(\mathbb{R}\).

• For a composite function \((f\circ g)(x) = f(g(x))\), the domain is \(\{ \text{domain of } f(g) \text{ in term of } x \}\) \(\cap\) \(\{ \text{domain of } g \text{ in term of } x \}\).

  For example, when \(f(x)=\frac{1}{x}\) and \(g(x) = ln(x)\)

  • the domain of \(f(g)\) in term of \(x\) is \(\{x : g(x) \neq 0\} = \{x : ln(x) \neq 0\} = \{x : x\neq 1\}\) and
  • the domain of \(g(x)\) is \(\{x : x > 0\}\), so
  • the domain of \(f\circ g\) is \(\{x : x > 0\} \cap \{x : x\neq 1\} = \{x : x > 0 \text{ and } x\neq 1\}\)

The resulting set of possible values of the dependent variable (\(y\)) is called the range.

• The range depends on the domain. For example,
  • the range of \(y = x+1, x\in (1,2)\) is \((2,3)\), while
  • the range of \(y = x+1, x\in (0,5)\) is \((1,6)\).
• If two functions have the same domain and the same rule, then they have the same range as well.

• \(f(x) = 1\)
• \(f(x) = x^2\)
• \(f(x) = \frac{1}{x+1}\)
• \(f(x) = \frac{1}{x^2 + 1}\)
• \(f(x) = \frac{1}{(x+1)^2}\)
• \(f(x) = \sqrt{x}\)
• \(f(x) = \sqrt{x}, x\in (9, 25]\)
• \(f(x) = ln(x-1)\)
• \(f(x) = 3^x\)

• Addition \(f(x) + g(x)\)
• Subtraction \(f(x) - g(x)\)
• Multiplication \(f(x) \cdot g(x)\)
• Division \(f(x)/g(x)\)

An exponential function with base \(a\) is defined as \(y = f(x) = a^x\) where \(a > 0\) and \(a\neq 1\)

• Suppose \(a > 0\) and \(a\neq 1\). Then
  • \(a^x \cdot a^y = a^{x+y}\)
  • \((a^x)^y = a^{xy}\)

  • (exponential equation) If \(a^x = a^y\), then \(x = y\).

    Example : Solve for \(x\) if \(9^x = 27\).

    • \(LHS = 9^x = (3^2)^x = 3^{2x}\) and
    • \(RHS = 27 = 3^3\), so
    • \(LHS = RHS\) implies \(2x = 3\) and thus
    • \(x=\frac{3}{2}\)

• change-of-base from \(a\) to \(e\)

  For any \(a > 0\), \(a^x = e^{(ln a)x}\)

If \(a > 0\) and \(a\neq 1\), then the logarithmic function of base \(a\) is defined by \(y = f(x) = log_a x\) for \(x > 0\).

• suppose \(a > 0, a\neq 1, x > 0, y > 0\), and \(r\) is any real number. Then
  • \(log_a x^r = r log_a x\)
  • \(log_a(xy) = log_a x + log_a y\)
  • \(log_a \frac{x}{y} = log_a(x\cdot y^{-1}) = log_a x + log_a y^{-1} = log_a x - log_a y\)
  • \(log_a a = 1\), and thus \(lne = 1\)
  • \(log_a 1 = 0\)
  • \(log_a a^r = r\)

  • (logarithmic equation) if \(log_a x = log_a y\), then \(x = y\)

    Example : Solve for \(x\) if \(3 ln (x) = 2 ln 8\).

    • \(LHS = 3ln(x) = ln(x^3)\) and
    • \(RHS = 2ln8 = ln8^2 = ln 64\), so
    • \(LHS=RHS\) implies \(x^3 = 64\) and thus
    • \(x=4\).

• change-of-base from \(a\) to \(e\)

  If \(a > 0, a\neq 1, b > 0\) and \(b\neq 1\), then for any \(x > 0\), \(log_a x = \frac{log_b x}{log_b a}\).

  Letting \(b=e\), then we have \(log_a x = \frac{ln(x)}{ln(a)}\).

• Identity

  For \(a > 0, a\neq 1\) and \(x > 0\), \(y = log_a x\) means \(a^y = x\).

  Thus we have \(y = log_a a^y\) and \(x = a^{log_a x}\) (equivalently, \(y = a^{log_a y}\)). When \(a=e\), we have \[ y = ln(e^y) = e^{lny} \]

• Rewrite a Log Expression

  \(ln \frac{\sqrt{x}e^x}{8x^3}\) \(= ln(\sqrt{x}e^x) - ln(8x^3)\) \(= ln(\sqrt{x}) + ln(e^x) - ln((2x)^3)\)

  \(= ln(x^{1/2}) + ln(e^x) - ln((2x)^3)\) \(= \frac{1}{2} ln(x) + x ln(e) - 3 ln(2x)\)

  \(= \frac{1}{2} ln(x) + x - 3 ln(2x)\)