This question tests your understanding that solving the equation f(x) = g(x) is equivalent to finding the x-coordinates where the graphs y = f(x) and y = g(x) intersect—a powerful visual and technological approach to solving equations. The intersection-solution connection works because at an intersection point, both functions have the same y-value: if the graphs meet at (a, b), then f(a) = b and g(a) = b, which means f(a) = g(a)—so x = a solves the equation f(x) = g(x)! For counting solutions to |x - 2| = x² - 4, we graph the V-shaped absolute value function y = |x - 2| (vertex at (2, 0), going up with slope ±1) and the parabola y = x² - 4 (vertex at (0, -4), opening upward). The absolute value function has two pieces: x - 2 for x ≥ 2 and -(x - 2) = 2 - x for x < 2. For x ≥ 2: x - 2 = x² - 4 gives x² - x - 2 = 0, so (x - 2)(x + 1) = 0, yielding x = 2 (valid since x ≥ 2). For x < 2: 2 - x = x² - 4 gives x² + x - 6 = 0, so (x + 3)(x - 2) = 0, yielding x = -3 (valid since x < 2). Choice C correctly identifies 2 solutions at x = -3 and x = 2. Choice A (0 solutions) would mean the graphs never intersect, Choice B (1 solution) misses one intersection, and Choice D (3 solutions) overcounts—perhaps confusing the number of pieces in the absolute value function with the number of intersections. The graphical approach makes counting solutions visual and intuitive: just count how many times the curves cross!

This question tests your understanding that solving the equation f(x) = g(x) is equivalent to finding the x-coordinates where the graphs y = f(x) and y = g(x) intersect—a powerful visual and technological approach to solving equations. The intersection-solution connection works because at an intersection point, both functions have the same y-value: if the graphs meet at (a, b), then f(a) = b and g(a) = b, which means f(a) = g(a)—so x = a solves the equation f(x) = g(x)! At an intersection (a, b), it means f(a) = g(a) = b. For an intersection at (4,7), f(4) = 7 and g(4) = 7, so f(4) = g(4). Choice B correctly states f(4) = g(4), as both equal 7 at x = 4. Choice D claims f(4) = g(7), but the point is (4,7), not relating to x=7—distinguish x and y coordinates carefully! The graphical solving recipe: (1) Graph both, (2) Note intersection (x,y), (3) Recognize f(x) = g(x) = y. Super progress—keep connecting points to functions!

This question tests your understanding that solving the equation f(x) = g(x) is equivalent to finding the x-coordinates where the graphs y = f(x) and y = g(x) intersect—a powerful visual and technological approach to solving equations. The intersection-solution connection works because at an intersection point, both functions have the same y-value: if the graphs meet at (a, b), then f(a) = b and g(a) = b, which means f(a) = g(a)—so x = a solves the equation f(x) = g(x)! This is why we can solve equations graphically: graph both sides as separate functions (y = left side and y = right side), find where they intersect, and read the x-coordinate(s). For solving f(x) = g(x) graphically, we graph both y = f(x) and y = g(x) on the same coordinate system, and the intersection points show where the functions are equal—for example, with f(x) = x² - 2 and g(x) = 2x + 1, they intersect at points with x-coordinates x = -1 and x = 3. Choice B correctly identifies the x-coordinates of intersections as approximately x ≈ -1.0 and x ≈ 3.0, which solve f(x) = g(x) because both functions equal the same y-value at those x-values. Choice D gives the y-coordinates instead of the x-coordinates from the intersection, but the solution to f(x) = g(x) is the x-value where they're equal, not the y-value they both equal—when reading intersections, trace carefully from the crossing point down to the x-axis and read the x-coordinate precisely! The graphical solving recipe: (1) Rewrite equation as f(x) = g(x), (2) Graph y = f(x) and y = g(x) on same axes, (3) Find intersection point(s) visually, (4) Read x-coordinate(s) of each intersection—those are your solutions, (5) Verify by substituting back. Keep practicing graphing to spot all intersections accurately—you've got this!

This question tests your understanding that solving the equation f(x) = g(x) is equivalent to finding the x-coordinates where the graphs y = f(x) and y = g(x) intersect—a powerful visual and technological approach to solving equations. The intersection-solution connection works because at an intersection point, both functions have the same y-value: if the graphs meet at (a, b), then f(a) = b and g(a) = b, which means f(a) = g(a)—so x = a solves the equation f(x) = g(x)! Technology makes this powerful for equations that are difficult to solve algebraically, like $2^x$ = 3x + 1, where we can use tables to approximate solutions by successive narrowing. Using tables, we can approximate by finding where f(x) and g(x) values are closest or where they cross (sign change in f(x) - g(x)), then narrow the interval with smaller steps—for $2^x$ = 3x + 1, the table shows a crossing between x = 3.5 and x = 3.6. Choice B correctly identifies x ≈ 3.5 as the best approximation to the nearest tenth, since at x = 3.5, the values are very close (11.3 ≈ 11.5), and the actual solution is nearby. Choice A claims x ≈ 3.0, but at x = 3.0, 8.0 < 10.0, and the crossing is later—count all points carefully and narrow intervals to avoid missing the precise spot! Table approximation method: (1) Create tables for both f(x) and g(x) at several x-values, (2) Look for where f(x) approximately equals g(x) or sign changes, (3) Narrow the interval with smaller steps, (4) Continue until desired precision. You're doing great—keep refining those approximations!

This question tests your understanding that solving the equation f(x) = g(x) is equivalent to finding the x-coordinates where the graphs y = f(x) and y = g(x) intersect—a powerful visual and technological approach to solving equations. The intersection-solution connection works because at an intersection point, both functions have the same y-value: if the graphs meet at (a, b), then f(a) = b and g(a) = b, which means f(a) = g(a)—so x = a solves the equation f(x) = g(x)! This is why we can solve equations graphically: graph both sides as separate functions (y = left side and y = right side), find where they intersect, and read the x-coordinate(s). Technology makes this powerful for equations that are difficult or impossible to solve algebraically, like $2^x$ = 3x + 1 or log(x) = $x^2$ - 5. Choice B correctly identifies the exact solutions x=-2, 0, 2 from the intersections. A tempting distractor like choice A uses wrong factors, perhaps misfactoring $x^3$ - 4x—remember to factor completely! The graphical solving recipe: graph $y=x^3$ and y=4x, find three crossings, verify—keep going, you've got this!

This question tests your understanding that solving the equation f(x) = g(x) is equivalent to finding the x-coordinates where the graphs y = f(x) and y = g(x) intersect—a powerful visual and technological approach to solving equations. The intersection-solution connection works because at an intersection point, both functions have the same y-value: if the graphs meet at (a, b), then f(a) = b and g(a) = b, which means f(a) = g(a)—so x = a solves the equation f(x) = g(x)! Using tables, we approximate solutions like $x^3$ = 2x + 5 by finding where values cross. The table shows a crossing between x = 2.0 (8 < 9) and x = 2.1 (9.261 > 9.2). Choice B correctly identifies x ≈ 2.1 as the best estimate to the nearest tenth, being closer to where they equal. Choice A claims x ≈ 2.0, but that's before the cross—narrow intervals to pinpoint precisely! Table approximation method: (1) Table values, (2) Find sign change, (3) Narrow with smaller steps, (4) Estimate. Fantastic effort—keep refining!

This question tests your understanding that solving the equation f(x) = g(x) is equivalent to finding the x-coordinates where the graphs y = f(x) and y = g(x) intersect—a powerful visual and technological approach to solving equations. The intersection-solution connection works because at an intersection point, both functions have the same y-value: if the graphs meet at (a, b), then f(a) = b and g(a) = b, which means f(a) = g(a)—so x = a solves the equation f(x) = g(x)! This is why we can solve equations graphically: graph both sides as separate functions (y = left side and y = right side), find where they intersect, and read the x-coordinate(s). Technology makes this powerful for equations that are difficult or impossible to solve algebraically, like $2^x$ = 3x + 1 or log(x) = $x^2$ - 5. Choice A correctly identifies that at an intersection, the y-values are equal, explaining why f(a)=g(a). A tempting distractor like choice B fails because slopes are not necessarily equal at intersections— that's for tangency, not equality. The transferable strategy: always remember the core idea that equal y-values at the same x mean the functions are equal there—great job grasping this!

This question tests your understanding that solving the equation f(x) = g(x) is equivalent to finding the x-coordinates where the graphs y = f(x) and y = g(x) intersect—a powerful visual and technological approach to solving equations. The intersection-solution connection works because at an intersection point, both functions have the same y-value: if the graphs meet at (a, b), then f(a) = b and g(a) = b, which means f(a) = g(a)—so x = a solves the equation f(x) = g(x)! This is why we can solve equations graphically: graph both sides as separate functions (y = left side and y = right side), find where they intersect, and read the x-coordinate(s). For (x+1)/(x-1) = 2, the graphs intersect at x=3, noting the vertical asymptote at x=1. Choice A correctly identifies x=3 as the solution. A tempting distractor like choice B might ignore the domain or mis-solve the equation—always solve algebraically and check. The transferable strategy: graph the rational function and horizontal line, find intersection away from asymptotes, verify— you're amazing!

This question tests your understanding that solving the equation f(x) = g(x) is equivalent to finding the x-coordinates where the graphs y = f(x) and y = g(x) intersect—a powerful visual and technological approach to solving equations. The intersection-solution connection works because at an intersection point, both functions have the same y-value: if the graphs meet at (a, b), then f(a) = b and g(a) = b, which means f(a) = g(a)—so x = a solves the equation f(x) = g(x)! The table shows that at x = 2.1, f(x) = $e^2$.1 ≈ 8.17 while g(x) = 4(2.1) = 8.40, so f(x) < g(x). At x = 2.2, f(x) = $e^2$.2 ≈ 9.03 while g(x) = 4(2.2) = 8.80, so f(x) > g(x). Since f(x) - g(x) changes from negative to positive between x = 2.1 and x = 2.2, the functions must be equal somewhere in this interval—this is the Intermediate Value Theorem in action! Choice A correctly identifies the interval [2.1, 2.2] where the sign change occurs. Choice B suggests [2.0, 2.1], but we'd need table values at x = 2.0 to confirm a sign change there. Choice C suggests [2.2, 2.3], but both functions would need to be evaluated at x = 2.3 to check. Choice D incorrectly claims no solution exists, missing the clear sign change in the table. The table approximation method uses sign changes in f(x) - g(x) to bracket solutions—when the difference switches from negative to positive (or vice versa), a solution lies between those x-values!

This question tests your understanding that solving the equation f(x) = g(x) is equivalent to finding the x-coordinates where the graphs y = f(x) and y = g(x) intersect—a powerful visual and technological approach to solving equations. The intersection-solution connection works because at an intersection point, both functions have the same y-value: if the graphs meet at (a, b), then f(a) = b and g(a) = b, which means f(a) = g(a)—so x = a solves the equation f(x) = g(x)! Technology makes this powerful for equations like ln(x) = x - 2, which are hard to solve algebraically. Using a graphing calculator, y = ln(x) and y = x - 2 intersect around x ≈ 3.15. Choice C correctly identifies x ≈ 3.15 to the nearest hundredth, as values are nearly equal there (ln(3.15) ≈ 1.147 ≈ 1.15). Choice D claims x ≈ 2.00, but at x = 2, ln(2) ≈ 0.69 > 0—check nearby points to narrow accurately! The graphical solving recipe: (1) Graph both, (2) Find intersection, (3) Use zoom or trace for precision, (4) Round as needed. You're mastering this—keep practicing!