Problem 35 - Algebra 2

Question

Identify each table of values as a type of function. A. \(\begin{array}{|c|c|}\hline x & {f(x)} \\ \hline-5 & {7} \\ \hline-3 & {5} \\ \hline-1 & {3} \\ \hline 0 & {2} \\ \hline 1 & {3} \\ \hline 3 & {5} \\ \hline 5 & {7} \\ \hline 7 & {9} \\ \hline\end{array}\) B. \(\begin{array}{|c|c|}\hline x & {f(x)} \\ \hline-5 & {24} \\ \hline-3 & {8} \\ \hline-1 & {0} \\ \hline 0 & {-1} \\ \hline 1 & {0} \\ \hline 3 & {8} \\ \hline 5 & {24} \\ \hline 7 & {48} \\ \hline\end{array}\) C. \(\begin{array}{|c|c|}\hline x & {f(x)} \\ \hline-1.3 & {-1} \\ \hline-1.7 & {-1} \\ \hline 0 & {1} \\ \hline 0.8 & {1} \\ \hline 0.9 & {1} \\ \hline 0.9 & {1} \\ \hline 1 & {2} \\ \hline 1.5 & {2} \\ \hline 2.3 & {3} \\\ \hline\end{array}\) D. \(\begin{array}{|c|c|}\hline x & {f(x)} \\ \hline-5 & {\text { undefined }} \\\ \hline-3 & {\text { undefined }} \\ \hline-1 & {\text { undefined }} \\\ \hline 0 & {0} \\ \hline 1 & {1} \\ \hline 4 & {2} \\ \hline 9 & {3} \\\ \hline 16 & {4} \\ \hline\end{array}\)

Step-by-Step Solution

Answer

A: Polynomial; B: Non-standard Polynomial; C: Piecewise/Step; D: Square Root.

Step 1: Recognize Patterns in Table A

Table A shows a set of values where, as each value of \(x\) increases or decreases in regular patterns, each change in \(f(x)\) also follows consistent increments or decrements about its neighbors. This can be assessed by observing the symmetry and repeated values in the function, indicating a mirror reflection across a vertical axis, typical of even functions like a polynomial or quadratic nature.

Step 2: Identify Patterns in Table B

In Table B, observe the symmetry across the \(y ext{-axis}\) in values of \(f(x)\) from \((1,0)\) onwards. It lacks the symmetrical increments or consistent change, distinct from popular trigonometric or squared patterns. Instead, this suggests points of evenness as reflections, which indicates a polynomial nature, but possibly non-standard.

Step 3: Determine Characteristics in Table C

Table C presents multiple occurrences where the \(f(x)\) values at different points are identical, showing clusters of output values despite unique inputs. This distinct non-consistent change coupled with repeated values indicates a step function, common in stair-step related functions or piecewise functions providing segment ranges for outputs.

Step 4: Analyze Table D for Function Type

In Table D, the early part shows undefined values, only beginning to produce outputs at perfect squares leading to stepwise responses aligned to square roots or logarithmic functions. This behavior is indicative of square root functions, which provide defined outputs only when the \(x\) value is a perfect square. Notice how these values progressively increase by square root integers.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions consisting of variables raised to various powers and combined using addition or subtraction. These functions can take on different forms, such as linear (degree 1), quadratic (degree 2), cubic (degree 3), etc.
The general format of a polynomial function is:

• \( f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \)

Each term adds a new dimension or 'factor' to the polynomial, often creating curves when plotted. The behavior of polynomial functions is fundamentally linear in relation to its degree, guiding a smooth, continuous plot without sudden jumps.
When you look at Table A in the exercise, the values of \( f(x) \) show a patterned increase with changes in \( x \), which can indicate an even polynomial function. The symmetry suggests reflection over the y-axis typical in quadratic functions, a subclass of polynomial functions.

Step Functions

Step functions are a kind of function where the function's value jumps from one level to another without any intermediate values in between. These functions often appear as series of flat horizontal lines on a graph, resembling steps.
An ideal way to think of step functions is by visualizing how floors in a building change with a staircase – you go up at one point, stay constant, then step up again. It's segmented and doesn't have slope variations.Table C exemplifies a step function through its repeated outputs for varying inputs. The function stays constant for specific ranges of \( x \) and "steps" up as \( x \) surpasses certain thresholds. This is common when functions are defined piecewise, meaning different equations govern different intervals of \( x \).
This piecewise structure provides clear boundaries making step functions unique in predictability across sections.

Even Functions

Even functions have a unique characteristic: they are symmetric about the y-axis. This means the function produces the same output for opposite inputs. Mathematically, an even function satisfies the equation:

• \( f(x) = f(-x) \)

The graphs of even functions are mirrored, connecting the points (x, y) and (-x, y).
In our example, Table A reflects this property, showing identical outputs for equidistant inputs from zero. This reflects an inherent symmetry commonly found in polynomials of even degree or other symmetrical functions.
Recognizing even functions is pivotal in solving equations, predicting values without exhaustive calculations, and understanding the inherent nature of various mathematical models seen in real-world phenomena.

Square Root Functions

Square root functions are functions that involve the square root of a variable. They have a distinct form:

• \( f(x) = \sqrt{x} \)

These functions generate values only for non-negative numbers because the square root of negative numbers is not real.
Graphs of square root functions depict a curve, beginning at the origin, moving right and slightly upward, showing growth that slows down as \( x \) increases.
When analyzing Table D, we notice values of \( f(x) \) only materialize when \( x \) is a perfect square, like 0, 1, 4, 9, etc. This is highly indicative of a square root function, as outputs only appear when it’s possible to take the square root of the number cleanly. These scenarios make square root functions essential in real-life for understanding gradual growth or decay patterns.