Rational Functions and Holes - AP Precalculus

Consider the function $g(x)=\dfrac{x^2-4}{x-2}$ . At which $x$ -value is $g$ undefined due to a removable discontinuity?
Answer: $x=2$ This question tests AP Precalculus skills: understanding of rational functions and identifying holes. A hole (removable discontinuity) occurs where both numerator and denominator share a common factor that equals zero. In this problem, g(x) = (x²-4)/(x-2) can be factored as g(x) = (x+2)(x-2)/(x-2), showing the common factor (x-2). Choice B is correct because at x = 2, both the numerator (2²-4 = 0) and denominator (2-2 = 0) equal zero, creating the removable discontinuity. Choice A is incorrect as it confuses the zero of the simplified function with the hole's location. To help students: Use the difference of squares factoring pattern. Remind them that after canceling, the hole remains at the canceled factor's zero.
Consider the function $u(x)=\dfrac{x^2-1}{x+1}$ for a simplified index; at which $x$ -value is it undefined due to a removable discontinuity?
Answer: $x=-1$ This question tests AP Precalculus skills: understanding of rational functions and identifying holes. A hole occurs where a rational function's numerator and denominator both equal zero, creating a removable discontinuity. In this problem, the function u(x) = (x²-1)/(x+1) can be factored as u(x) = (x+1)(x-1)/(x+1), showing a common factor of (x+1). Choice B is correct because when x = -1, both the numerator (1-1=0) and denominator (-1+1=0) equal zero, and the (x+1) factor cancels, creating a hole at x = -1. Choice A is incorrect because x = 1 only makes the numerator zero after factoring, not the denominator. To help students: Recognize x²-1 as a difference of squares. Pay attention to the sign in the denominator (x+1) to identify the correct hole location.
Consider the function $v(x)=\dfrac{x^2-10x+25}{x-5}$ for a demand adjustment; which value of $x$ makes numerator and denominator zero, creating a hole?
Answer: $x=5$ This question tests AP Precalculus skills: understanding of rational functions and identifying holes. A hole occurs where a rational function's numerator and denominator both equal zero, creating a removable discontinuity. In this problem, the function v(x) = (x²-10x+25)/(x-5) has a numerator that factors as (x-5)², so v(x) = (x-5)²/(x-5), with a common factor of (x-5). Choice D is correct because when x = 5, both the numerator (25-50+25=0) and denominator (5-5=0) equal zero, and one (x-5) factor cancels, creating a hole at x = 5. Choice B is incorrect because x = 10 doesn't make either the numerator or denominator zero. To help students: Recognize perfect square trinomials like x²-10x+25 = (x-5)². Practice the pattern (x-a)² = x²-2ax+a² for quick factoring.
Consider the function $q(x)=\dfrac{x^2-2x-3}{x-3}$ for a water-flow model; which value of $x$ makes the numerator and denominator zero, creating a hole?
Answer: $x=3$ This question tests AP Precalculus skills: understanding of rational functions and identifying holes. A hole occurs where a rational function's numerator and denominator both equal zero, creating a removable discontinuity. In this problem, the function q(x) = (x²-2x-3)/(x-3) can be factored as q(x) = (x-3)(x+1)/(x-3), revealing a common factor of (x-3). Choice B is correct because when x = 3, both the numerator (9-6-3=0) and denominator (3-3=0) equal zero, and the (x-3) factor cancels out, creating a hole at x = 3. Choice D is incorrect because x = -3 doesn't make either the numerator or denominator zero. To help students: Use factoring techniques for trinomials where the leading coefficient is 1. Check your factorization by expanding to verify it matches the original expression.
Consider the function $p(x)=\dfrac{x^2+x}{x}$ for a per-item cost model; at which $x$ -value is it undefined due to a removable discontinuity?
Answer: $x=0$ This question tests AP Precalculus skills: understanding of rational functions and identifying holes. A hole occurs where a rational function's numerator and denominator both equal zero, creating a removable discontinuity. In this problem, the function p(x) = (x²+x)/x can be factored as p(x) = x(x+1)/x, showing a common factor of x. Choice C is correct because when x = 0, both the numerator (0+0=0) and denominator (0) equal zero, and the x factor cancels, creating a hole at x = 0. Choice A is incorrect because x = 1 doesn't make the denominator zero. To help students: Factor out common terms from the numerator before simplifying. Remind students that division by zero creates discontinuities, but when both parts are zero, it's removable.
Consider the function $t(x)=\dfrac{x^2-6x+9}{x-3}$ for a revenue model; which value of $x$ makes the numerator and denominator zero, creating a hole?
Answer: $x=3$ This question tests AP Precalculus skills: understanding of rational functions and identifying holes. A hole occurs where a rational function's numerator and denominator both equal zero, creating a removable discontinuity. In this problem, the function t(x) = (x²-6x+9)/(x-3) can be recognized as t(x) = (x-3)²/(x-3), since x²-6x+9 is a perfect square trinomial, showing a common factor of (x-3). Choice B is correct because when x = 3, both the numerator (9-18+9=0) and denominator (3-3=0) equal zero, and one (x-3) factor cancels out, creating a hole at x = 3. Choice D is incorrect as x = 6 doesn't make either expression zero. To help students: Recognize perfect square trinomial patterns (a-b)² = a²-2ab+b². Note that even with repeated factors, only one cancels with the denominator to create the hole.
Consider the function $q(x)=\dfrac{x^2-6x+9}{x-3}$ for a scaling rule; what is the location of the hole?
Answer: $x=3$ This question tests AP Precalculus skills: understanding of rational functions and identifying holes. A hole occurs where a rational function's numerator and denominator both equal zero, creating a removable discontinuity. In this problem, the function q(x) = (x²-6x+9)/(x-3) has a numerator that factors as (x-3)², so q(x) = (x-3)²/(x-3), with a common factor of (x-3). Choice B is correct because when x = 3, both the numerator (9-18+9=0) and denominator (3-3=0) equal zero, and one (x-3) factor cancels, creating a hole at x = 3. Choice A is incorrect because x = -3 doesn't make either part zero. To help students: Recognize perfect square trinomials like x²-6x+9 = (x-3)². Practice factoring quadratics to identify common factors with the denominator.
Consider the function $s(x)=\dfrac{x^2+x-6}{x-2}$ for a demand curve; at which $x$ -value is $s$ undefined due to a removable discontinuity?
Answer: $x=2$ This question tests AP Precalculus skills: understanding of rational functions and identifying holes. A hole occurs where a rational function's numerator and denominator both equal zero, creating a removable discontinuity. In this problem, the function s(x) = (x²+x-6)/(x-2) can be factored as s(x) = (x-2)(x+3)/(x-2), showing a common factor of (x-2). Choice B is correct because when x = 2, both the numerator (4+2-6=0) and denominator (2-2=0) equal zero, and the (x-2) factor cancels out, creating a hole at x = 2. Choice A is incorrect because x = -3 only makes the numerator zero after cancellation, not creating a hole. To help students: Factor trinomials by finding two numbers that multiply to give the constant term and add to give the middle coefficient. Always verify by substituting the x-value into both numerator and denominator.
Consider $h(x)=\dfrac{x^2+5x+6}{x+2}$ . Which value of $x$ makes numerator and denominator zero, creating a hole?
Answer: $x=-2$ This question tests AP Precalculus skills: understanding of rational functions and identifying holes. A hole occurs where a rational function's numerator and denominator both equal zero, creating a removable discontinuity. In this problem, h(x) = (x²+5x+6)/(x+2) requires factoring the numerator: x²+5x+6 = (x+2)(x+3), revealing the common factor (x+2). Choice A is correct because when x = -2, both the numerator (4-10+6 = 0) and denominator (-2+2 = 0) equal zero, confirming the hole's location. Choice B is incorrect as it identifies where the simplified function h(x) = x+3 equals zero. To help students: Practice factoring quadratic expressions. Emphasize checking that both original numerator and denominator equal zero at the proposed hole location.
Consider the function $p(x)=\dfrac{x^2+5x+6}{x+2}$ for a production estimate; at which $x$ -value is $p$ undefined due to a removable discontinuity?
Answer: $x=-2$ This question tests AP Precalculus skills: understanding of rational functions and identifying holes. A hole occurs where a rational function's numerator and denominator both equal zero, creating a removable discontinuity. In this problem, the function p(x) = (x²+5x+6)/(x+2) can be factored as p(x) = (x+2)(x+3)/(x+2), showing a common factor of (x+2). Choice A is correct because when x = -2, both the numerator (4-10+6=0) and denominator (-2+2=0) equal zero, and the (x+2) factor cancels out, creating a hole at x = -2. Choice B is incorrect because x = -3 only makes the numerator zero after cancellation, representing a zero of the simplified function. To help students: Factor trinomials systematically to find factors like (x+2)(x+3). Always verify holes by checking that both original numerator and denominator equal zero.
Consider the function $v(x)=\dfrac{x^2+3x}{x}$ for a unit-rate calculation; at which $x$ -value is $v$ undefined due to a removable discontinuity?
Answer: $x=0$ This question tests AP Precalculus skills: understanding of rational functions and identifying holes. A hole occurs where a rational function's numerator and denominator both equal zero, creating a removable discontinuity. In this problem, the function v(x) = (x²+3x)/x can be factored as v(x) = x(x+3)/x, showing a common factor of x. Choice C is correct because when x = 0, both the numerator (0+0=0) and denominator (0) equal zero, and the x factor cancels out, creating a hole at x = 0. Choice A is incorrect because x = -3 only makes the numerator zero after cancellation, not creating a hole. To help students: Factor out common terms from the numerator before identifying cancelable factors. Be careful with x = 0 cases, as they often create holes when x appears in both numerator and denominator.
Consider the function $f(x)=\dfrac{x^2-1}{x-1}$ for a ticket-pricing model; what is the location of the hole?
Answer: $x=1$ This question tests AP Precalculus skills: understanding of rational functions and identifying holes. A hole occurs where a rational function's numerator and denominator both equal zero, creating a removable discontinuity. In this problem, the function f(x) = (x²-1)/(x-1) can be factored as f(x) = (x+1)(x-1)/(x-1), revealing a common factor of (x-1). Choice C is correct because when x = 1, both the numerator and denominator equal zero, and the (x-1) factor cancels out, creating a hole at x = 1. Choice B is incorrect because x = 5 doesn't make either the numerator or denominator zero. To help students: Emphasize factoring the numerator first to identify common factors with the denominator. Practice recognizing difference of squares patterns like x²-1 = (x+1)(x-1).
Consider the function $g(x)=\dfrac{x^2-4}{x-2}$ for average fuel cost; at which $x$ -value is it undefined due to a removable discontinuity?
Answer: $x=2$ This question tests AP Precalculus skills: understanding of rational functions and identifying holes. A hole occurs where a rational function's numerator and denominator both equal zero, creating a removable discontinuity. In this problem, the function g(x) = (x²-4)/(x-2) can be factored as g(x) = (x+2)(x-2)/(x-2), showing a common factor of (x-2). Choice B is correct because at x = 2, both the numerator (4-4=0) and denominator (2-2=0) equal zero, and the (x-2) factor cancels, creating a hole. Choice A is incorrect because while x = -2 makes the numerator zero, it doesn't make the denominator zero. To help students: Practice factoring difference of squares (x²-4 = (x+2)(x-2)). Emphasize that holes require both numerator AND denominator to be zero at the same x-value.
Consider the function $f(x)=\dfrac{x^2-1}{x-1}$ . What is the location of the hole?
Answer: $x=1$ This question tests AP Precalculus skills: understanding of rational functions and identifying holes. A hole occurs where a rational function's numerator and denominator both equal zero, creating a removable discontinuity. In this problem, the function f(x) = (x²-1)/(x-1) can be factored as f(x) = (x+1)(x-1)/(x-1), revealing a common factor of (x-1) that cancels. Choice B is correct because when x = 1, both the original numerator (1²-1 = 0) and denominator (1-1 = 0) equal zero, confirming the hole's location. Choice A is incorrect because it identifies where the simplified function equals zero, not where the hole occurs. To help students: Emphasize factoring the numerator completely before canceling. Practice identifying the difference between zeros of the simplified function and locations of holes.
Consider the function $h(x)=\dfrac{x^2-1}{x-1}$ for a sensor calibration; what is the location of the hole in the function $h(x)$ ?
Answer: $x=1$ This question tests AP Precalculus skills: understanding of rational functions and identifying holes. A hole occurs where a rational function's numerator and denominator both equal zero, creating a removable discontinuity. In this problem, the function h(x) = (x²-1)/(x-1) can be factored as h(x) = (x+1)(x-1)/(x-1), revealing a common factor of (x-1). Choice B is correct because when x = 1, both the numerator (1-1=0) and denominator (1-1=0) equal zero, and the (x-1) factor cancels out, creating a hole at x = 1. Choice A is incorrect because x = -1 only makes the numerator zero, which creates a zero of the simplified function, not a hole. To help students: Remember the difference of squares factoring pattern (a²-b² = (a+b)(a-b)). Practice identifying and canceling common factors to find removable discontinuities.
Consider the function $h(x)=\dfrac{x^2-9}{x-3}$ for a speed ratio; which value of $x$ makes numerator and denominator zero, creating a hole?
Answer: $x=3$ This question tests AP Precalculus skills: understanding of rational functions and identifying holes. A hole occurs where a rational function's numerator and denominator both equal zero, creating a removable discontinuity. In this problem, the function h(x) = (x²-9)/(x-3) can be factored as h(x) = (x+3)(x-3)/(x-3), revealing a common factor of (x-3). Choice D is correct because when x = 3, both the numerator (9-9=0) and denominator (3-3=0) equal zero, and the (x-3) factor cancels out, creating a hole at x = 3. Choice A is incorrect because x = -3 only makes the numerator zero, not the denominator. To help students: Recognize x²-9 as a difference of squares that factors to (x+3)(x-3). Always check that the suspected hole location makes BOTH parts of the fraction equal zero.
Consider the function $f(x)=\dfrac{x^2-9}{x-3}$ for a shipping-rate model; at which $x$ -value is $f$ undefined due to a removable discontinuity?
Answer: $x=3$ This question tests AP Precalculus skills: understanding of rational functions and identifying holes. A hole occurs where a rational function's numerator and denominator both equal zero, creating a removable discontinuity. In this problem, the function f(x) = (x²-9)/(x-3) can be factored as f(x) = (x+3)(x-3)/(x-3), revealing a common factor of (x-3). Choice B is correct because when x = 3, both the numerator and denominator equal zero, and the (x-3) factor cancels out, creating a hole at x = 3. Choice A is incorrect because x = -3 makes only the numerator zero, resulting in a zero of the function, not a hole. To help students: Emphasize that holes require both numerator AND denominator to be zero. Practice factoring quadratics to identify common factors that create removable discontinuities.
Consider the function $g(x)=\dfrac{x^2-4x}{x-4}$ for a pricing rule; which value of $x$ makes the numerator and denominator zero, creating a hole?
Answer: $x=4$ This question tests AP Precalculus skills: understanding of rational functions and identifying holes. A hole occurs where a rational function's numerator and denominator both equal zero, creating a removable discontinuity. In this problem, the function g(x) = (x²-4x)/(x-4) can be factored as g(x) = x(x-4)/(x-4), showing a common factor of (x-4). Choice A is correct because when x = 4, both the numerator (16-16=0) and denominator (4-4=0) equal zero, and the (x-4) factor cancels out, creating a hole at x = 4. Choice B is incorrect because x = 0 only makes the numerator zero, resulting in a zero of the function rather than a hole. To help students: Factor the numerator completely before identifying common factors. Use substitution to verify that both numerator and denominator equal zero at the hole location.
Consider $n(x)=\dfrac{x^2+x-12}{x+4}$ . What is the location of the hole in $n(x)$ ?
Answer: $x=-4$ This question tests AP Precalculus skills: understanding of rational functions and identifying holes. A hole occurs where a rational function's numerator and denominator both equal zero, creating a removable discontinuity. In this problem, n(x) = (x²+x-12)/(x+4) requires factoring the numerator: x²+x-12 = (x+4)(x-3), revealing the common factor (x+4). Choice B is correct because when x = -4, both the numerator (16-4-12 = 0) and denominator (-4+4 = 0) equal zero, creating the hole. Choice A is incorrect as it identifies where the simplified function n(x) = x-3 equals zero, not the hole's location. To help students: Use systematic factoring to find (x+4)(x-3) from x²+x-12. Remind them to always verify by substituting the proposed hole location into both parts of the original fraction.
Consider the function $r(x)=\dfrac{x^2-2x}{x}$ for a rate calculation; which value of $x$ makes numerator and denominator zero, creating a hole?
Answer: $x=0$ This question tests AP Precalculus skills: understanding of rational functions and identifying holes. A hole occurs where a rational function's numerator and denominator both equal zero, creating a removable discontinuity. In this problem, the function r(x) = (x²-2x)/x can be factored as r(x) = x(x-2)/x, revealing a common factor of x. Choice C is correct because when x = 0, both the numerator (0-0=0) and denominator (0) equal zero, and the x factor cancels out, creating a hole at x = 0. Choice A is incorrect because x = 2 only makes the numerator zero after factoring, not the denominator. To help students: Always factor the numerator completely before identifying common factors. Emphasize that x = 0 can create holes just like any other value.

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