Algebra 2 Flashcards: Using Intersections To Solve Equivalent Functions

What this deck covers

This deck focuses on Using Intersections To Solve Equivalent Functions, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra 2.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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