Combine Different Function Types - Algebra
Card 1 of 30
What is $(f+g)(2)$ if $f(2)=7$ and $g(2)=-3$?
What is $(f+g)(2)$ if $f(2)=7$ and $g(2)=-3$?
Tap to reveal answer
$4$. Add the function values: $7+(-3)=4$.
$4$. Add the function values: $7+(-3)=4$.
← Didn't Know|Knew It →
For $T(t)=70+20\left(\frac{1}{2}\right)^t$, what is the ambient temperature?
For $T(t)=70+20\left(\frac{1}{2}\right)^t$, what is the ambient temperature?
Tap to reveal answer
$70$. The constant term represents the ambient temperature.
$70$. The constant term represents the ambient temperature.
← Didn't Know|Knew It →
Find $T(0)$ for $T(t)=70+20\left(\frac{1}{2}\right)^t$.
Find $T(0)$ for $T(t)=70+20\left(\frac{1}{2}\right)^t$.
Tap to reveal answer
$90$. Substitute $t=0$: $70+20\left(\frac{1}{2}\right)^0=70+20(1)=90$.
$90$. Substitute $t=0$: $70+20\left(\frac{1}{2}\right)^0=70+20(1)=90$.
← Didn't Know|Knew It →
Simplify $(\frac{f}{g})(x)$ if $f(x)=10^x$ and $g(x)=2^x$.
Simplify $(\frac{f}{g})(x)$ if $f(x)=10^x$ and $g(x)=2^x$.
Tap to reveal answer
$(\frac{f}{g})(x)=5^x$. Use property $\left(\frac{a}{b}\right)^x=\frac{a^x}{b^x}$: $\frac{10^x}{2^x}=\left(\frac{10}{2}\right)^x=5^x$.
$(\frac{f}{g})(x)=5^x$. Use property $\left(\frac{a}{b}\right)^x=\frac{a^x}{b^x}$: $\frac{10^x}{2^x}=\left(\frac{10}{2}\right)^x=5^x$.
← Didn't Know|Knew It →
What is the vertical shift of $f(x)$ when forming $f(x)+c$?
What is the vertical shift of $f(x)$ when forming $f(x)+c$?
Tap to reveal answer
Shift up by $c$ units. Positive $c$ moves the graph upward by $c$ units.
Shift up by $c$ units. Positive $c$ moves the graph upward by $c$ units.
← Didn't Know|Knew It →
For $f(x)=ab^x$, which condition on $b$ gives exponential growth?
For $f(x)=ab^x$, which condition on $b$ gives exponential growth?
Tap to reveal answer
$b>1$. Base greater than 1 creates increasing exponential behavior.
$b>1$. Base greater than 1 creates increasing exponential behavior.
← Didn't Know|Knew It →
For $f(x)=ab^x$, which condition on $b$ gives exponential decay?
For $f(x)=ab^x$, which condition on $b$ gives exponential decay?
Tap to reveal answer
$0<b<1$. Base between 0 and 1 creates decreasing exponential behavior.
$0<b<1$. Base between 0 and 1 creates decreasing exponential behavior.
← Didn't Know|Knew It →
What is the difference between $(f+g)(x)$ and $f(x)+g(x)$?
What is the difference between $(f+g)(x)$ and $f(x)+g(x)$?
Tap to reveal answer
$(f+g)(x)=f(x)+g(x)$. They are identical - both notations represent the same operation.
$(f+g)(x)=f(x)+g(x)$. They are identical - both notations represent the same operation.
← Didn't Know|Knew It →
What is the standard form of an exponential function used in Algebra $1$: growth or decay?
What is the standard form of an exponential function used in Algebra $1$: growth or decay?
Tap to reveal answer
$f(x)=ab^x$. Standard exponential form with base $b$ and initial value $a$.
$f(x)=ab^x$. Standard exponential form with base $b$ and initial value $a$.
← Didn't Know|Knew It →
If $T(t)=A+Be^{-kt}$ with $k>0$, what is $\lim_{t\to\infty}T(t)$?
If $T(t)=A+Be^{-kt}$ with $k>0$, what is $\lim_{t\to\infty}T(t)$?
Tap to reveal answer
$A$. The exponential term approaches zero as $t$ approaches infinity.
$A$. The exponential term approaches zero as $t$ approaches infinity.
← Didn't Know|Knew It →
In $T(t)=A+Be^{-kt}$, what is the initial temperature $T(0)$ in terms of $A$ and $B$?
In $T(t)=A+Be^{-kt}$, what is the initial temperature $T(0)$ in terms of $A$ and $B$?
Tap to reveal answer
$T(0)=A+B$. Substitute $t=0$ into the expression and simplify.
$T(0)=A+B$. Substitute $t=0$ into the expression and simplify.
← Didn't Know|Knew It →
In $T(t)=A+Be^{-kt}$ for cooling, what does the parameter $k>0$ control?
In $T(t)=A+Be^{-kt}$ for cooling, what does the parameter $k>0$ control?
Tap to reveal answer
Cooling rate (how fast it approaches $A$). Larger $k$ means faster cooling toward the ambient temperature.
Cooling rate (how fast it approaches $A$). Larger $k$ means faster cooling toward the ambient temperature.
← Didn't Know|Knew It →
What is the vertical scaling of $f(x)$ when forming $cf(x)$?
What is the vertical scaling of $f(x)$ when forming $cf(x)$?
Tap to reveal answer
Multiply all outputs by $c$. Each output value is multiplied by the constant $c$.
Multiply all outputs by $c$. Each output value is multiplied by the constant $c$.
← Didn't Know|Knew It →
In $T(t)=A+Be^{-kt}$ for cooling, what does the constant $A$ represent?
In $T(t)=A+Be^{-kt}$ for cooling, what does the constant $A$ represent?
Tap to reveal answer
Ambient (surrounding) temperature. The final temperature the object approaches as time goes to infinity.
Ambient (surrounding) temperature. The final temperature the object approaches as time goes to infinity.
← Didn't Know|Knew It →
Find $h(x)$ if $f(x)=\sqrt{x}$ and $g(x)=x$ and $h(x)=f(x)g(x)$.
Find $h(x)$ if $f(x)=\sqrt{x}$ and $g(x)=x$ and $h(x)=f(x)g(x)$.
Tap to reveal answer
$h(x)=x\sqrt{x}$. Multiply the expressions: $\sqrt{x}\cdot x=x\sqrt{x}$.
$h(x)=x\sqrt{x}$. Multiply the expressions: $\sqrt{x}\cdot x=x\sqrt{x}$.
← Didn't Know|Knew It →
What is the domain of $h(x)=x\sqrt{x}$?
What is the domain of $h(x)=x\sqrt{x}$?
Tap to reveal answer
$x\ge 0$. Square root function requires $x\ge 0$.
$x\ge 0$. Square root function requires $x\ge 0$.
← Didn't Know|Knew It →
Identify the error: $(f+g)(x)=f(x)\cdot g(x)$. What is the correct statement?
Identify the error: $(f+g)(x)=f(x)\cdot g(x)$. What is the correct statement?
Tap to reveal answer
$(f+g)(x)=f(x)+g(x)$. Addition of functions uses $+$, not multiplication ($\cdot$).
$(f+g)(x)=f(x)+g(x)$. Addition of functions uses $+$, not multiplication ($\cdot$).
← Didn't Know|Knew It →
Identify the error: $(\frac{f}{g})(x)$ has domain $\text{Dom}(f)\cap\text{Dom}(g)$. What is missing?
Identify the error: $(\frac{f}{g})(x)$ has domain $\text{Dom}(f)\cap\text{Dom}(g)$. What is missing?
Tap to reveal answer
Also require $g(x)\ne 0$. Division requires the additional restriction that $g(x)\ne 0$.
Also require $g(x)\ne 0$. Division requires the additional restriction that $g(x)\ne 0$.
← Didn't Know|Knew It →
What function type is $T(t)=A+Be^{-kt}$ when $k>0$?
What function type is $T(t)=A+Be^{-kt}$ when $k>0$?
Tap to reveal answer
Constant plus decaying exponential. Models cooling behavior approaching a constant temperature $A$.
Constant plus decaying exponential. Models cooling behavior approaching a constant temperature $A$.
← Didn't Know|Knew It →
Simplify $(fg)(x)$ if $f(x)=2^x$ and $g(x)=3^x$.
Simplify $(fg)(x)$ if $f(x)=2^x$ and $g(x)=3^x$.
Tap to reveal answer
$(fg)(x)=6^x$. Use property $(ab)^x=a^x b^x$: $2^x\cdot 3^x=(2\cdot 3)^x=6^x$.
$(fg)(x)=6^x$. Use property $(ab)^x=a^x b^x$: $2^x\cdot 3^x=(2\cdot 3)^x=6^x$.
← Didn't Know|Knew It →
Simplify $(f-g)(x)$ if $f(x)=5^x$ and $g(x)=2\cdot 5^x$.
Simplify $(f-g)(x)$ if $f(x)=5^x$ and $g(x)=2\cdot 5^x$.
Tap to reveal answer
$(f-g)(x)=-1\cdot 5^x$. Factor out $5^x$: $5^x-2\cdot 5^x=-1\cdot 5^x$.
$(f-g)(x)=-1\cdot 5^x$. Factor out $5^x$: $5^x-2\cdot 5^x=-1\cdot 5^x$.
← Didn't Know|Knew It →
Simplify $(f+g)(x)$ if $f(x)=3^x$ and $g(x)=2\cdot 3^x$.
Simplify $(f+g)(x)$ if $f(x)=3^x$ and $g(x)=2\cdot 3^x$.
Tap to reveal answer
$(f+g)(x)=3\cdot 3^x$. Factor out $3^x$: $3^x+2\cdot 3^x=3\cdot 3^x$.
$(f+g)(x)=3\cdot 3^x$. Factor out $3^x$: $3^x+2\cdot 3^x=3\cdot 3^x$.
← Didn't Know|Knew It →
Find the domain of $h(x)=\frac{2^x}{x^2-9}$.
Find the domain of $h(x)=\frac{2^x}{x^2-9}$.
Tap to reveal answer
All real $x$ except $x=-3$ and $x=3$. Denominator $x^2-9=(x-3)(x+3)$ cannot equal zero.
All real $x$ except $x=-3$ and $x=3$. Denominator $x^2-9=(x-3)(x+3)$ cannot equal zero.
← Didn't Know|Knew It →
What is the domain of $h(x)=\sqrt{x-3}+2^x$?
What is the domain of $h(x)=\sqrt{x-3}+2^x$?
Tap to reveal answer
$x\ge 3$. Square root requires $x-3\ge 0$, so $x\ge 3$.
$x\ge 3$. Square root requires $x-3\ge 0$, so $x\ge 3$.
← Didn't Know|Knew It →
What is the domain of $h(x)=\sqrt{x}+\frac{1}{x}$?
What is the domain of $h(x)=\sqrt{x}+\frac{1}{x}$?
Tap to reveal answer
$x>0$. Both square root and fraction require $x>0$.
$x>0$. Both square root and fraction require $x>0$.
← Didn't Know|Knew It →
What is the domain of $h(x)=\frac{x+1}{x-2}$?
What is the domain of $h(x)=\frac{x+1}{x-2}$?
Tap to reveal answer
All real $x$ except $x=2$. Denominator cannot be zero, so exclude $x=2$.
All real $x$ except $x=2$. Denominator cannot be zero, so exclude $x=2$.
← Didn't Know|Knew It →
What is the definition of $(f-g)(x)$ in terms of $f(x)$ and $g(x)$?
What is the definition of $(f-g)(x)$ in terms of $f(x)$ and $g(x)$?
Tap to reveal answer
$(f-g)(x)=f(x)-g(x)$. Subtract the second function's output from the first at each input.
$(f-g)(x)=f(x)-g(x)$. Subtract the second function's output from the first at each input.
← Didn't Know|Knew It →
Find $(f-g)(x)$ if $f(x)=x^2+4$ and $g(x)=3x$.
Find $(f-g)(x)$ if $f(x)=x^2+4$ and $g(x)=3x$.
Tap to reveal answer
$(f-g)(x)=x^2-3x+4$. Subtract the expressions: $(x^2+4)-3x=x^2-3x+4$.
$(f-g)(x)=x^2-3x+4$. Subtract the expressions: $(x^2+4)-3x=x^2-3x+4$.
← Didn't Know|Knew It →
Find $(f+g)(x)$ if $f(x)=2x-1$ and $g(x)=x^2$.
Find $(f+g)(x)$ if $f(x)=2x-1$ and $g(x)=x^2$.
Tap to reveal answer
$(f+g)(x)=x^2+2x-1$. Add the expressions: $(2x-1)+x^2=x^2+2x-1$.
$(f+g)(x)=x^2+2x-1$. Add the expressions: $(2x-1)+x^2=x^2+2x-1$.
← Didn't Know|Knew It →
What is $(\frac{f}{g})(4)$ if $f(4)=10$ and $g(4)=-5$?
What is $(\frac{f}{g})(4)$ if $f(4)=10$ and $g(4)=-5$?
Tap to reveal answer
$-2$. Divide the function values: $\frac{10}{-5}=-2$.
$-2$. Divide the function values: $\frac{10}{-5}=-2$.
← Didn't Know|Knew It →