Find the maximal domain of an algebraic function

What is the first method for finding the domain of the displayed function?

Function and interval

\(\displaystyle f(x)=(x-a)^2-b^2,\quad x\in[a-2b,a+2b]\)

Test the claim that positive endpoint values prevent interior roots. Inspect the function over the interval below. Change a or b while keeping b positive. This table gives exact theoretical values

Why is every displayed setting a counterexample to the claim?

Maximal domain

The maximal domain is the largest set of real inputs for which every operation in the function is defined. Check each denominator and radicand separately, then intersect the resulting sets

1. 1Exclude denominator zeros A denominator must not equal zero \(\displaystyle \frac{1}{x-4}:x-4\ne0\Rightarrow x\ne4\)
2. 2Include numerator root endpoints A square-root radicand in the numerator may equal zero \(\displaystyle \sqrt{x-2}:x-2\ge0\Rightarrow x\ge2\)
3. 3Make denominator radicands positive A square root in a denominator cannot have a zero radicand \(\displaystyle \frac{1}{\sqrt{x+1}}:x+1>0\Rightarrow x>-1\)
4. 4Intersect all component domains An input must satisfy every restriction in a sum or difference \(\displaystyle [2,\infty)\cap\left(\mathbb{R}\setminus\{5\}\right)=[2,5)\cup(5,\infty)\)

Given

\(\displaystyle \sqrt{2x-6},\quad 2x-6=0\)

Solve the boundary equation for the numerator square root

Given

\(\displaystyle \sqrt{2x-6},\quad x=3,\quad 2(3)-6\)

Evaluate the radicand at the boundary to check its endpoint status

A numerator square-root endpoint is made strict. One of these lines is right. Which one?

Keep an actual technology record

Use technology with graphical, numerical and symbolic functions. Keep the entered expression, assumptions, selected domain and viewing window, actual output, independent hand check and explanation. External execution and written work are ungraded; the checked entries assess individual mathematical components

1. 1Use the functions the task needs Use a CAS for the symbolic solving step, not just a numerical solver or a graph. If your tool has no symbolic facility, leave that execution step pending until a suitable tool is available. Hand algebra verifies an output; it does not stand in for a recorded technology run
2. 2Separate three kinds of bounds The mathematical domain determines allowed inputs. A deliberate domain restriction changes the problem. A viewing window only controls the portion displayed and may conceal valid features. Record the displayed horizontal and vertical limits separately

Worked model: refute a root claim

Claim: a continuous function with positive values at both ends of an interval has no roots inside. A counterexample must meet the stated conditions and contradict the conclusion. Use the function and interval below

1. 1Plan the technology record Enter (x-0.5)^2-1 with explicit brackets. Use x from −1.5 to 2.5 and y from −1.5 to 3.5. Request a table at −1.5, −0.5, 0.5, 1.5 and 2.5. These settings reveal the endpoints, vertex and crossings
2. 2Expected output to check The corresponding exact values are 3, 0, −1, 0, 3. This is an authored prediction, not a claim that a particular device was run
3. 3Model explanation The polynomial is continuous, and both endpoint values are positive. Its two roots are strictly inside the interval, so the claim is false. The graph helps find the example; the substitutions below verify it \(\displaystyle \begin{aligned}f(-\frac32)&=(-2)^2-1=3,&f(\frac52)&=2^2-1=3,\\f(-\frac12)&=(-1)^2-1=0,&f(\frac32)&=1^2-1=0\end{aligned}\)

Your task: produce and verify a counterexample

Choose your own a from −1, 0, 1, 2 and b from 0.5, 1, 1.5. Use your calculator, spreadsheet or graphing tool to produce a graph and five-row table for f(x) = (x − a)² − b² on [a − 2b,a + 2b]. Choose a vertical window that shows −b² and 3b². Save three settings, changing only a or only b at a time. Compare their graphs and tables and explain what changes and what stays true. Keep your actual input and output; do not substitute the authored table for your technology record

1. 1Written explanation State the claim, your parameter choices, continuity, both endpoint signs and the two interior roots. Explain why one valid counterexample refutes a universal claim. A graph picture alone is not that explanation
2. 2Independent checks Substitute the endpoints and roots by hand. Check that b is positive, the roots are strictly inside and every table value matches your entered rule. If the output differs, investigate brackets or settings before concluding

Given

\(\displaystyle f(x)=(x+1)^2-9/4,\quad x=-4\)

Separate check: for a = −1 and b = 3/2, calculate f at the left endpoint a − 2b

Given

\(\displaystyle a=-1,\quad b=3/2\)

For this same fixed example, calculate the smaller root a − b.

Worked model: preserve meaning across notation

The formal rule and its original restriction are shown below. A common linear input is (x^2-16)/(x-4). The ^ denotes a power and the brackets group the entire numerator and denominator. Your device may require different syntax; record its actual input and output

1. 1Simplified output and original domain Factoring and cancellation give the result below only on the original domain. A simplifier may display the polynomial without the exclusion. The original expression remains undefined at 4; the displayed polynomial alone would give 8 there \(\displaystyle \frac{(x-4)(x+4)}{x-4}=x+4,\quad x\ne4\)
2. 2An output convention worth checking For real x, sqrt(x^2) is |x|, often written abs(x), not always x. At x = −3 the value is 3. With an assumption x ≥ 0 a tool can legitimately simplify it to x; record assumptions as well as output
3. 3Model comparison At x = 2 both admissible forms of r give 6, but agreement at one input does not prove identity. Factorisation proves agreement on the original domain. Domain restrictions and real-variable assumptions are part of the statement

Your task: translate and compare technology output

Choose a = 2, 3 or 5 and enter the fraction below into technology using its supported syntax. Use symbolic simplification and record its actual output, then compare a table of the original and candidate x + a at admissible inputs and at x = a. Also inspect sqrt(x²) at a negative and positive input

1. 1Make a three-column record Write the formal expression and domain, your exact technology input/settings, and the actual output. Explain each power, bracket, absolute-value notation and any undefined/error result. Record real-variable assumptions or restrictions you supplied
2. 2Model for the reasoning The two rational forms agree when x ≠ a because the common factor can be cancelled there. The polynomial extends to a value at x = a, while the original does not. For sqrt(x²), a negative test distinguishes |x| from x; factorisation or a sign argument completes the explanation

Given

\(\displaystyle r(x)=\frac{x^2-25}{x-5},\quad x=-2\)

Separate check: evaluate the original rational expression at x = −2.

For the original rule below, a tool displays x + 5. What must you preserve?

Given

\(\displaystyle \sqrt{x^2},\quad x=-7/2\)

Evaluate the principal square root at this negative input

Worked plan: exact roots, decimal values and a graph

Take the rule below on the real domain and solve for its zeros. Enter brackets around the translated input. Symbolic solving should give the exact roots shown below. Numerical evaluation gives approximately −0.4142135624 and 2.4142135624. A graph with x from −1 to 3 and y from −3 to 3 shows the vertex and both crossings. The table at −1,0,1,2,3 gives 2,−1,−2,−1,2. These are authored expected results, not a claim that a device was run

1. 1Solve and check the exact roots Rearrange the equation, then substitute either exact root into the original rule \(\displaystyle \begin{aligned}(x-1)^2-2&=0\\(x-1)^2&=2\\x-1&=\pm\sqrt2\\x&=1\pm\sqrt2\\f(1\pm\sqrt2)&=(\pm\sqrt2)^2-2=0\end{aligned}\)
2. 2Why a narrow graph can mislead A horizontal window from 0.5 to 1.5 hides both roots or intersections. That display does not prove they do not exist. Restore the larger window and use the exact symbolic answers and independent substitution to explain the difference
3. 3Certify rounding separately The exact squares show 1.4142135 < √2 < 1.4142136. Translate both endpoints for the larger root and reverse them when subtracting for the smaller root. Both endpoints of each resulting interval round to the same three-decimal value, certifying −0.414 and 2.414. A displayed midpoint or truncated screen alone is not this proof

Rule

\(\displaystyle f(x)=(x-1)^2-2\)

For the worked polynomial, find f(3)

Produce graphical, symbolic and numerical results

Choose one of the parameter values below. Use your tool’s symbolic solver to find the zeros and keep its exact output. Evaluate every solution numerically, retain the unrounded values and report three decimal places. Independently solve the equation by hand and substitute the exact roots into the original rule

1. 1Choose and justify a useful view Produce a graph and a table including a−2,a−1,a,a+1,a+2. Choose horizontal and vertical window limits that reveal the vertex and both roots. Save your actual settings/output and explain why they are suitable. Then use the narrow horizontal window [a−0.5,a+0.5] and record which roots disappear. Finally apply the domain restriction [a−2,a+2] in the tool and save that graph. State its true range and compare it with the unrestricted range. Changing a view and restricting the domain are different operations
2. 2Verify, explain and retain Use 1.4142135 < √2 < 1.4142136 to certify both rounded answers by endpoint rounding, not just by copying the display. Check the entered brackets, variable value, exact solution form and table values. Compare the usefulness and limitations of the three functions you used. Keep the complete run record and your independent reasoning for review

Review every part of the record

For every permitted a, the exact roots are a−√2 and a+√2. Their three-decimal values are (0.586,3.414), (1.586,4.414) or (2.586,5.414) for a=2,3,4 respectively. The table is 2,−1,−2,−1,2, the vertex is (a,−2), and both roots lie inside [a−2,a+2]. The narrow window hides both roots because √2>0.5. The unrestricted range is [−2,∞), while the restricted domain [a−2,a+2] gives range [−2,2]. Confirm that the exact symbolic expression, the decimal approximations and the actual graph describe the same two roots

1. 1Compare the actual record with the prediction The displayed numbers and model graph are checks, not evidence of your run. Compare your actual commands, output, assumptions, selected domain and window, and independent proof. Investigate missing branches or roots before accepting the result