Using Intersections to Solve Equivalent Functions - Algebra 2

1. Identify the intersection $x$-value: $f(x)=-x+5$ and $g(x)=2x-1$.

   $x=2$. Set $-x+5=2x-1$ and solve: $6=3x$, so $x=2$.

2. Find the intersection $x$-values: $f(x)=\frac{1}{x-1}$ and $g(x)=1$.

   $x=2$. Set $\frac{1}{x-1}=1$ and solve: $1=x-1$, so $x=2$.

3. What is the domain restriction when solving $\log(x)$ equations by intersections?

   Require $x>0$ (argument must be positive). Logarithms are undefined for non-positive arguments.

4. Identify the intersection $x$-value: $f(x)=3x-2$ and $g(x)=x+6$.

   $x=4$. Set $3x-2=x+6$ and solve: $2x=8$, so $x=4$.

5. Identify the intersection $x$-value: $f(x)=2x+1$ and $g(x)=7$.

   $x=3$. Set $2x+1=7$ and solve: $2x=6$, so $x=3$.

6. What does a tangent intersection point correspond to for $h(x)=f(x)-g(x)$?

   A zero where $h(x)$ touches $0$ without sign change. The difference function touches zero but doesn't cross.

7. Find the intersection $x$-value: $f(x)=e^x$ and $g(x)=1$.

   $x=0$. Set $e^x=1=e^0$, so $x=0$.

8. What does it mean if $y=f(x)$ and $y=g(x)$ overlap completely?

   Infinitely many solutions (all $x$ in the domain). Identical graphs mean every x satisfies the equation.

9. What does it mean if $f(x)-g(x)>0$ at a given $x$?

   $f(x)>g(x)$ at that $x$. Positive difference means $f$ is above $g$.

10. Find the intersection $x$-value: $f(x)=\ln(x)$ and $g(x)=\ln(5)$.

    $x=5$. Set $\ln(x)=\ln(5)$: $x=5$.

11. Find the intersection $x$-value: $f(x)=\ln(x)$ and $g(x)=1$.

    $x=e$. Set $\ln(x)=1$: $x=e^1=e$.

12. What does it mean if $f(x)-g(x)<0$ at a given $x$?

    $f(x)<g(x)$ at that $x$. Negative difference means $f$ is below $g$.

13. Identify the valid solution of $\ln(x-2)=0$: $x=2$ or $x=3$?

    $x=3$. Only $x=3$ makes the argument $x-2=1>0$ valid.

14. Find the intersection $x$-value: $f(x)=\log_2(x)$ and $g(x)=5$.

    $x=32$. Set $\log_2(x)=5$: $x=2^5=32$.

15. Find the intersection $x$-value: $f(x)=\ln(x-2)$ and $g(x)=0$.

    $x=3$. Set $\ln(x-2)=0$: $x-2=1$, so $x=3$.

16. Which $x$-values are solutions of $f(x)=g(x)$: the intersection $x$-coordinates or $y$-coordinates?

    The intersection $x$-coordinates. Solutions are x-values where graphs meet, not y-values.

17. What is the key step after solving an equation formed by a rational intersection?

    Check for extraneous solutions from $0$ denominators. Solutions making denominators zero are invalid.

18. Find the intersection $x$-values: $f(x)=\frac{1}{x}$ and $g(x)=-1$.

    $x=-1$. Set $\frac{1}{x}=-1$ and solve: $1=-x$, so $x=-1$.

19. Find the intersection $x$-values: $f(x)=x^2+2x$ and $g(x)=0$.

    $x=-2$ and $x=0$. Set $x^2+2x=0$ and factor: $x(x+2)=0$.

20. What approximate method uses repeated narrowing of an interval where a sign change occurs?

    Bisection (successive approximations). Repeatedly halve intervals where sign changes occur.

21. Find the intersection $x$-value: $f(x)=3^x$ and $g(x)=27$.

    $x=3$. Set $3^x=27=3^3$, so $x=3$.

22. Find the intersection $x$-values: $f(x)=\frac{x}{x-1}$ and $g(x)=2$.

    $x=2$. Set $\frac{x}{x-1}=2$: $x=2(x-1)$, so $x=2$.

23. What is the standard method to turn a graph-intersection problem into an equation?

    Set $f(x)=g(x)$ and solve for $x$. This creates an equation with one variable to solve.

24. What is the domain restriction when solving $\log(x-3)$ equations by intersections?

    Require $x-3>0$, so $x>3$. The argument $x-3$ must be positive for the log to exist.

25. What is the intersection $x$-value if $f(x)=g(x)$ occurs at point $(4,-3)$?

    $x=4$. Intersection coordinates give the solution x-value directly.

26. What function's zeros correspond to intersections of $y=f(x)$ and $y=g(x)$?

    Zeros of $h(x)=f(x)-g(x)$. When $h(x)=0$, the original functions are equal.

27. What is the key step after solving an equation formed by a logarithmic intersection?

    Check that all log arguments are positive. Solutions making log arguments non-positive are invalid.

28. Find the intersection $x$-value: $f(x)=2^x$ and $g(x)=8$.

    $x=3$. Set $2^x=8=2^3$, so $x=3$.

29. What coordinate(s) solve $f(x)=g(x)$ when the graphs intersect at $(a,b)$?

    $x=a$. Only the x-coordinate is needed to solve the equation.

30. What is the typical graphical output used to approximate solutions to $f(x)=g(x)$?

    The intersection point(s) of the two graphs. Graph intersections visually show where functions are equal.

31. What is the intersection $x$-value of $f(x)=x^3$ and $g(x)=0$?

    $x=0$. Set $x^3=0$, so $x=0$.

32. Find the intersection $x$-value(s): $f(x)=x^3$ and $g(x)=x$.

    $x=-1$, $x=0$, and $x=1$. Set $x^3=x$ and factor: $x(x^2-1)=x(x-1)(x+1)=0$.

33. Which interval brackets a solution if $h(1)=-2$ and $h(2)=3$ for $h(x)=f(x)-g(x)$?

    A solution lies in $(1,2)$. Sign change from negative to positive indicates a zero crossing.

34. Find the intersection $x$-values: $f(x)=|x|$ and $g(x)=2$.

    $x=-2$ and $x=2$. Set $|x|=2$: $x=2$ or $x=-2$.

35. Find the intersection $x$-value(s): $f(x)=x^2$ and $g(x)=-x^2$.

    $x=0$. Set $x^2=-x^2$: $2x^2=0$, so $x=0$.

36. Which midpoint does bisection test first on the interval $[2,6]$?

    $x=4$. Bisection starts at the midpoint of the interval.

37. Identify the intersection $x$-value: $f(x)=x$ and $g(x)=\sqrt{x}$.

    $x=0$ and $x=1$. Set $x=\sqrt{x}$ and square: $x^2=x$, so $x(x-1)=0$.

38. What does it mean if the graphs of $y=f(x)$ and $y=g(x)$ do not intersect?

    $f(x)=g(x)$ has no real solutions. Non-intersecting graphs means the equation has no solution.

39. What are the intersection $x$-values of $f(x)=x^2$ and $g(x)=x$?

    $x=0$ and $x=1$. Set $x^2=x$ and solve: $x^2-x=0$, so $x(x-1)=0$.

40. Find the intersection $x$-value: $f(x)=\log(x)$ and $g(x)=2$.

    $x=100$. Set $\log(x)=2$: $x=10^2=100$.

41. Identify the excluded value when solving $\frac{x+1}{x-3}=2$ by intersections.

    Exclude $x=3$. The denominator $x-3$ cannot equal zero.

42. What are the intersection $x$-values of $f(x)=x^2-1$ and $g(x)=0$?

    $x=-1$ and $x=1$. Set $x^2-1=0$ and factor: $(x-1)(x+1)=0$.

43. What table-based clue suggests an intersection between $x=a$ and $x=b$?

    $f(a)-g(a)$ and $f(b)-g(b)$ have opposite signs. Opposite signs of $f-g$ indicate a crossing between points.

44. What is the domain restriction when solving rational intersections like $\frac{1}{x-2}=g(x)$?

    Require $x\ne 2$. Rational functions are undefined when denominators equal zero.

45. Find the intersection $x$-value: $f(x)=10^x$ and $g(x)=1000$.

    $x=3$. Set $10^x=1000=10^3$, so $x=3$.

46. Find the intersection $x$-values: $f(x)=\frac{1}{x}$ and $g(x)=1$.

    $x=1$. Set $\frac{1}{x}=1$ and solve: $1=x$.

47. Find the intersection $x$-values: $f(x)=x^2-4x$ and $g(x)=0$.

    $x=0$ and $x=4$. Set $x^2-4x=0$ and factor: $x(x-4)=0$.

48. Find the intersection $x$-values: $f(x)=\frac{2}{x}$ and $g(x)=1$.

    $x=2$. Set $\frac{2}{x}=1$ and solve: $2=x$.

49. Find the intersection $x$-values: $f(x)=|x-1|$ and $g(x)=3$.

    $x=-2$ and $x=4$. Set $|x-1|=3$: $x-1=3$ or $x-1=-3$.

50. What is the key step after solving an equation formed by an absolute value intersection?

    Verify solutions satisfy the original absolute value equation. Absolute value equations can introduce extraneous solutions.

51. Find the intersection $x$-value: $f(x)=\ln(x)$ and $g(x)=0$.

    $x=1$. Set $\ln(x)=0$: $x=e^0=1$.

52. Find the intersection $x$-value(s): $f(x)=|x|$ and $g(x)=-x$.

    $x\le 0$. When $x\leq 0$, $|x|=-x$, so they're equal.

53. Find the intersection $x$-value(s): $f(x)=|x|$ and $g(x)=x$.

    $x\ge 0$. When $x\geq 0$, $|x|=x$, so they're equal.

54. What are the intersection $x$-values of $f(x)=x^2$ and $g(x)=4$?

    $x=-2$ and $x=2$. Set $x^2=4$ and solve: $x=\pm\sqrt{4}=\pm 2$.

55. What conclusion can you make if $f(x)-g(x)$ changes sign from $x=a$ to $x=b$?

    There is at least one solution in $(a,b)$. Sign changes indicate the graphs cross (Intermediate Value Theorem).

56. What equation must be true at an intersection of $y=f(x)$ and $y=g(x)$?

    $f(x)=g(x)$. At intersections, both functions have equal y-values.

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