Using Intersections to Solve Equivalent Functions - Algebra 2
Card 1 of 30
Find the intersection $x$-value: $f(x)=3^x$ and $g(x)=27$.
Find the intersection $x$-value: $f(x)=3^x$ and $g(x)=27$.
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$x=3$. Set $3^x=27=3^3$, so $x=3$.
$x=3$. Set $3^x=27=3^3$, so $x=3$.
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What is the intersection $x$-value of $f(x)=x^3$ and $g(x)=0$?
What is the intersection $x$-value of $f(x)=x^3$ and $g(x)=0$?
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$x=0$. Set $x^3=0$, so $x=0$.
$x=0$. Set $x^3=0$, so $x=0$.
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What is the intersection $x$-value if $f(x)=g(x)$ occurs at point $(4,-3)$?
What is the intersection $x$-value if $f(x)=g(x)$ occurs at point $(4,-3)$?
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$x=4$. Intersection coordinates give the solution x-value directly.
$x=4$. Intersection coordinates give the solution x-value directly.
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Which midpoint does bisection test first on the interval $[2,6]$?
Which midpoint does bisection test first on the interval $[2,6]$?
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$x=4$. Bisection starts at the midpoint of the interval.
$x=4$. Bisection starts at the midpoint of the interval.
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Which interval brackets a solution if $h(1)=-2$ and $h(2)=3$ for $h(x)=f(x)-g(x)$?
Which interval brackets a solution if $h(1)=-2$ and $h(2)=3$ for $h(x)=f(x)-g(x)$?
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A solution lies in $(1,2)$. Sign change from negative to positive indicates a zero crossing.
A solution lies in $(1,2)$. Sign change from negative to positive indicates a zero crossing.
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Find the intersection $x$-value(s): $f(x)=x^2$ and $g(x)=-x^2$.
Find the intersection $x$-value(s): $f(x)=x^2$ and $g(x)=-x^2$.
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$x=0$. Set $x^2=-x^2$: $2x^2=0$, so $x=0$.
$x=0$. Set $x^2=-x^2$: $2x^2=0$, so $x=0$.
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Find the intersection $x$-value(s): $f(x)=x^3$ and $g(x)=x$.
Find the intersection $x$-value(s): $f(x)=x^3$ and $g(x)=x$.
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$x=-1$, $x=0$, and $x=1$. Set $x^3=x$ and factor: $x(x^2-1)=x(x-1)(x+1)=0$.
$x=-1$, $x=0$, and $x=1$. Set $x^3=x$ and factor: $x(x^2-1)=x(x-1)(x+1)=0$.
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Find the intersection $x$-value: $f(x)=\ln(x)$ and $g(x)=\ln(5)$.
Find the intersection $x$-value: $f(x)=\ln(x)$ and $g(x)=\ln(5)$.
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$x=5$. Set $\ln(x)=\ln(5)$: $x=5$.
$x=5$. Set $\ln(x)=\ln(5)$: $x=5$.
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Identify the intersection $x$-value: $f(x)=x$ and $g(x)=\sqrt{x}$.
Identify the intersection $x$-value: $f(x)=x$ and $g(x)=\sqrt{x}$.
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$x=0$ and $x=1$. Set $x=\sqrt{x}$ and square: $x^2=x$, so $x(x-1)=0$.
$x=0$ and $x=1$. Set $x=\sqrt{x}$ and square: $x^2=x$, so $x(x-1)=0$.
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Identify the valid solution of $\ln(x-2)=0$: $x=2$ or $x=3$?
Identify the valid solution of $\ln(x-2)=0$: $x=2$ or $x=3$?
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$x=3$. Only $x=3$ makes the argument $x-2=1>0$ valid.
$x=3$. Only $x=3$ makes the argument $x-2=1>0$ valid.
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Find the intersection $x$-value: $f(x)=\ln(x-2)$ and $g(x)=0$.
Find the intersection $x$-value: $f(x)=\ln(x-2)$ and $g(x)=0$.
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$x=3$. Set $\ln(x-2)=0$: $x-2=1$, so $x=3$.
$x=3$. Set $\ln(x-2)=0$: $x-2=1$, so $x=3$.
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Find the intersection $x$-value: $f(x)=10^x$ and $g(x)=1000$.
Find the intersection $x$-value: $f(x)=10^x$ and $g(x)=1000$.
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$x=3$. Set $10^x=1000=10^3$, so $x=3$.
$x=3$. Set $10^x=1000=10^3$, so $x=3$.
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Find the intersection $x$-value: $f(x)=e^x$ and $g(x)=1$.
Find the intersection $x$-value: $f(x)=e^x$ and $g(x)=1$.
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$x=0$. Set $e^x=1=e^0$, so $x=0$.
$x=0$. Set $e^x=1=e^0$, so $x=0$.
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Find the intersection $x$-value: $f(x)=\ln(x)$ and $g(x)=0$.
Find the intersection $x$-value: $f(x)=\ln(x)$ and $g(x)=0$.
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$x=1$. Set $\ln(x)=0$: $x=e^0=1$.
$x=1$. Set $\ln(x)=0$: $x=e^0=1$.
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Find the intersection $x$-value: $f(x)=\log(x)$ and $g(x)=2$.
Find the intersection $x$-value: $f(x)=\log(x)$ and $g(x)=2$.
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$x=100$. Set $\log(x)=2$: $x=10^2=100$.
$x=100$. Set $\log(x)=2$: $x=10^2=100$.
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Find the intersection $x$-value: $f(x)=\log_2(x)$ and $g(x)=5$.
Find the intersection $x$-value: $f(x)=\log_2(x)$ and $g(x)=5$.
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$x=32$. Set $\log_2(x)=5$: $x=2^5=32$.
$x=32$. Set $\log_2(x)=5$: $x=2^5=32$.
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Find the intersection $x$-value: $f(x)=\ln(x)$ and $g(x)=1$.
Find the intersection $x$-value: $f(x)=\ln(x)$ and $g(x)=1$.
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$x=e$. Set $\ln(x)=1$: $x=e^1=e$.
$x=e$. Set $\ln(x)=1$: $x=e^1=e$.
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What is the typical graphical output used to approximate solutions to $f(x)=g(x)$?
What is the typical graphical output used to approximate solutions to $f(x)=g(x)$?
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The intersection point(s) of the two graphs. Graph intersections visually show where functions are equal.
The intersection point(s) of the two graphs. Graph intersections visually show where functions are equal.
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Identify the intersection $x$-value: $f(x)=2x+1$ and $g(x)=7$.
Identify the intersection $x$-value: $f(x)=2x+1$ and $g(x)=7$.
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$x=3$. Set $2x+1=7$ and solve: $2x=6$, so $x=3$.
$x=3$. Set $2x+1=7$ and solve: $2x=6$, so $x=3$.
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Identify the intersection $x$-value: $f(x)=3x-2$ and $g(x)=x+6$.
Identify the intersection $x$-value: $f(x)=3x-2$ and $g(x)=x+6$.
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$x=4$. Set $3x-2=x+6$ and solve: $2x=8$, so $x=4$.
$x=4$. Set $3x-2=x+6$ and solve: $2x=8$, so $x=4$.
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Identify the intersection $x$-value: $f(x)=-x+5$ and $g(x)=2x-1$.
Identify the intersection $x$-value: $f(x)=-x+5$ and $g(x)=2x-1$.
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$x=2$. Set $-x+5=2x-1$ and solve: $6=3x$, so $x=2$.
$x=2$. Set $-x+5=2x-1$ and solve: $6=3x$, so $x=2$.
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What are the intersection $x$-values of $f(x)=x^2$ and $g(x)=4$?
What are the intersection $x$-values of $f(x)=x^2$ and $g(x)=4$?
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$x=-2$ and $x=2$. Set $x^2=4$ and solve: $x=\pm\sqrt{4}=\pm 2$.
$x=-2$ and $x=2$. Set $x^2=4$ and solve: $x=\pm\sqrt{4}=\pm 2$.
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What are the intersection $x$-values of $f(x)=x^2-1$ and $g(x)=0$?
What are the intersection $x$-values of $f(x)=x^2-1$ and $g(x)=0$?
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$x=-1$ and $x=1$. Set $x^2-1=0$ and factor: $(x-1)(x+1)=0$.
$x=-1$ and $x=1$. Set $x^2-1=0$ and factor: $(x-1)(x+1)=0$.
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What are the intersection $x$-values of $f(x)=x^2$ and $g(x)=x$?
What are the intersection $x$-values of $f(x)=x^2$ and $g(x)=x$?
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$x=0$ and $x=1$. Set $x^2=x$ and solve: $x^2-x=0$, so $x(x-1)=0$.
$x=0$ and $x=1$. Set $x^2=x$ and solve: $x^2-x=0$, so $x(x-1)=0$.
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Find the intersection $x$-values: $f(x)=x^2+2x$ and $g(x)=0$.
Find the intersection $x$-values: $f(x)=x^2+2x$ and $g(x)=0$.
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$x=-2$ and $x=0$. Set $x^2+2x=0$ and factor: $x(x+2)=0$.
$x=-2$ and $x=0$. Set $x^2+2x=0$ and factor: $x(x+2)=0$.
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Find the intersection $x$-values: $f(x)=x^2-4x$ and $g(x)=0$.
Find the intersection $x$-values: $f(x)=x^2-4x$ and $g(x)=0$.
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$x=0$ and $x=4$. Set $x^2-4x=0$ and factor: $x(x-4)=0$.
$x=0$ and $x=4$. Set $x^2-4x=0$ and factor: $x(x-4)=0$.
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Find the intersection $x$-values: $f(x)=|x|$ and $g(x)=2$.
Find the intersection $x$-values: $f(x)=|x|$ and $g(x)=2$.
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$x=-2$ and $x=2$. Set $|x|=2$: $x=2$ or $x=-2$.
$x=-2$ and $x=2$. Set $|x|=2$: $x=2$ or $x=-2$.
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Find the intersection $x$-values: $f(x)=|x-1|$ and $g(x)=3$.
Find the intersection $x$-values: $f(x)=|x-1|$ and $g(x)=3$.
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$x=-2$ and $x=4$. Set $|x-1|=3$: $x-1=3$ or $x-1=-3$.
$x=-2$ and $x=4$. Set $|x-1|=3$: $x-1=3$ or $x-1=-3$.
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Find the intersection $x$-value(s): $f(x)=|x|$ and $g(x)=x$.
Find the intersection $x$-value(s): $f(x)=|x|$ and $g(x)=x$.
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$x\ge 0$. When $x\geq 0$, $|x|=x$, so they're equal.
$x\ge 0$. When $x\geq 0$, $|x|=x$, so they're equal.
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Find the intersection $x$-value(s): $f(x)=|x|$ and $g(x)=-x$.
Find the intersection $x$-value(s): $f(x)=|x|$ and $g(x)=-x$.
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$x\le 0$. When $x\leq 0$, $|x|=-x$, so they're equal.
$x\le 0$. When $x\leq 0$, $|x|=-x$, so they're equal.
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