9.3 Functions and General Mathematics

Why Functions Matter on the USTET

Philippine Senior High School General Mathematics puts functions at the center of the curriculum: linear, quadratic, piecewise, rational, exponential, and logarithmic models. The USTET Mathematics subtest draws on this Grade 11 background. You will not write long proofs, but you must evaluate f(x), read a table, state a domain restriction, and occasionally interpret a growth model such as compound interest or population. This section builds the language and the fastest exam moves.

What Is a Function?

A function is a rule that pairs each input from a set (the domain) with exactly one output in another set (the range). Think of a vendo machine: each button code should release one specific item. If the same input could produce two different outputs, the relation is not a function.

Vertical Line Test

If any vertical line intersects a graph more than once, the graph is not a function of x. A sideways parabola x = y² fails; an ordinary upright parabola y = x² passes.

Function Notation

We write y = f(x), read "f of x." The symbol f(x) is not multiplication of f and x. It means "the output of function f when the input is x."

If f(x) = 2x² − 3x + 1, then:

• f(3) = 2(9) − 3(3) + 1 = 18 − 9 + 1 = 10
• f(−1) = 2(1) − 3(−1) + 1 = 2 + 3 + 1 = 6
• f(a + 1) = 2(a + 1)² − 3(a + 1) + 1 — expand only if the question asks for a simplified polynomial

Worked Example 1: Evaluate from a Rule

Given g(x) = (x − 4)/(x + 2), find g(6).

g(6) = (6 − 4)/(6 + 2) = 2/8 = 1/4. Note that x = −2 is excluded from the domain because the denominator would be zero.

Worked Example 2: Read from a Table

Then h(1) = −1 and h(−2) = 5. If a stem asks for h(h(0)), first h(0) = 1, then h(1) = −1.

Domain and Range

Domain = allowable inputs. Common restrictions on USTET-style items:

• Denominator ≠ 0 for rational expressions
• Expression under an even root ≥ 0
• Argument of a real logarithm > 0

Range = possible outputs. For f(x) = x², domain is all real numbers and range is y ≥ 0. For a linear function f(x) = mx + b with m ≠ 0, both domain and range are all real numbers.

Worked Example 3: Domain of a Rational Function

Find the domain of f(x) = (x + 1)/(x² − 9).

Factor: x² − 9 = (x − 3)(x + 3). Exclude x = 3 and x = −3. Domain: all real x except −3 and 3.

Linear and Piecewise Functions

A linear function has the form f(x) = mx + b, where m is the slope (rate of change) and b is the y-intercept. Slope between two points: m = (y₂ − y₁)/(x₂ − x₁).

A piecewise function uses different rules on different intervals. Example:

f(x) = { x + 2 if x < 0; x² if x ≥ 0 }

Then f(−3) = −3 + 2 = −1, and f(3) = 9. Always check which piece owns the input — especially at the boundary x = 0 here, which uses x².

Operations and Composition (Entrance Level)

• (f + g)(x) = f(x) + g(x)
• (f · g)(x) = f(x) · g(x)
• Composition (f ∘ g)(x) = f(g(x)) — apply g first, then f

If f(x) = 2x + 1 and g(x) = x², then (f ∘ g)(3) = f(9) = 19, while (g ∘ f)(3) = g(7) = 49. Order matters.

Inverses (What USTET Expects)

The inverse f⁻¹ undoes f: f⁻¹(f(x)) = x. To find an inverse algebraically for a simple linear rule:

1. Replace f(x) with y.
2. Swap x and y.
3. Solve for y; that expression is f⁻¹(x).

Example: f(x) = 3x − 6 → y = 3x − 6 → x = 3y − 6 → x + 6 = 3y → y = (x + 6)/3. So f⁻¹(x) = (x + 6)/3. Check: f(4) = 6 and f⁻¹(6) = 4.

Note: f⁻¹ does not mean 1/f(x).

Exponential Snapshot from Gen Math

An exponential function f(x) = a · bˣ (a ≠ 0, b > 0, b ≠ 1) models growth when b > 1 and decay when 0 < b < 1. Simple compound-interest style: A = P(1 + r)ᵗ.

If PHP 5,000 grows at 4% per year for 2 years with annual compounding: A = 5000(1.04)² = 5000(1.0816) = PHP 5,408. USTET may ask for the amount after t years or for which expression models the situation — focus on identifying base (1 + r) and exponent t.

Logarithms undo exponentials: if bʸ = x then y = log_b(x). At entrance depth, know that log₁₀(1000) = 3 and that log_b(bᵏ) = k.

Common Traps

• Reading f(2) as "f times 2" instead of evaluating at 2.
• Treating f⁻¹(x) as 1/f(x).
• Forgetting domain holes where a denominator is zero.
• Using the wrong piece of a piecewise function at a boundary point.
• Swapping composition order: f(g(x)) is not automatically g(f(x)).

Section Takeaway

A function gives one output per valid input. Evaluate with careful substitution, state domains by excluding illegal inputs, and read tables and graphs with the vertical line test in mind. Know composition order, the meaning of an inverse, and the basic exponential model A = P(1 + r)ᵗ. These Gen Math ideas are high-yield for the USTET Mathematics subtest.