Algebra Flashcards: Combine Different Function Types

What this deck covers

This deck focuses on Combine Different Function Types, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra.

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: Find $h(x)$ if $f(x)=\sqrt{x}$ and $g(x)=x$ and $h(x)=f(x)g(x)$.

Answer: $h(x)=x\sqrt{x}$. Multiply the expressions: $\sqrt{x}\cdot x=x\sqrt{x}$.

Flashcard 2: Simplify $(f+g)(x)$ if $f(x)=3^x$ and $g(x)=2\cdot 3^x$.

Answer: $(f+g)(x)=3\cdot 3^x$. Factor out $3^x$: $3^x+2\cdot 3^x=3\cdot 3^x$.

Flashcard 3: What is the vertical scaling of $f(x)$ when forming $cf(x)$?

Answer: Multiply all outputs by $c$. Each output value is multiplied by the constant $c$.

Flashcard 4: Identify the exponential part in $h(x)=4+2^x$.

Answer: $2^x$. The exponential function has the form $a\cdot b^x$.

Flashcard 5: What is $(f+g)(2)$ if $f(2)=7$ and $g(2)=-3$?

Answer: $4$. Add the function values: $7+(-3)=4$.

Flashcard 6: Find $(\frac{f}{g})(x)$ if $f(x)=x+1$ and $g(x)=x-2$.

Answer: $(\frac{f}{g})(x)=\frac{x+1}{x-2}$. Form the quotient of the two expressions.

Flashcard 7: What is the domain of $(f+g)(x)$ in terms of the domains of $f$ and $g$?

Answer: $\text{Dom}(f+g)=\text{Dom}(f)\cap\text{Dom}(g)$. Domain is the intersection of both individual domains.

Flashcard 8: What is the domain of $h(x)=x\sqrt{x}$?

Answer: $x\ge 0$. Square root function requires $x\ge 0$.

Flashcard 9: If $T(t)=A+Be^{-kt}$ with $k>0$, what is $\lim_{t\to\infty}T(t)$?

Answer: $A$. The exponential term approaches zero as $t$ approaches infinity.

Flashcard 10: What is the result of adding a constant $c$ to a function $f(x)$?

Answer: $f(x)+c$. Adding a constant shifts the function vertically by $c$ units.

Flashcard 11: What function type is $T(t)=A+Be^{-kt}$ when $k>0$?

Answer: Constant plus decaying exponential. Models cooling behavior approaching a constant temperature $A$.

Flashcard 12: What is $(f-g)(-1)$ if $f(-1)=5$ and $g(-1)=12$?

Answer: $-7$. Subtract the function values: $5-12=-7$.

Flashcard 13: Find $h(x)$ if $f(x)=x^2-1$ and $g(x)=3$ and $h(x)=f(x)+g(x)$.

Answer: $h(x)=x^2+2$. Add the expressions: $(x^2-1)+3=x^2+2$.

Flashcard 14: What is $(\frac{f}{g})(4)$ if $f(4)=10$ and $g(4)=-5$?

Answer: $-2$. Divide the function values: $\frac{10}{-5}=-2$.

Flashcard 15: Find $h(x)$ if $f(x)=\sqrt{x}$ and $g(x)=x$ and $h(x)=f(x)g(x)$.

Answer: $h(x)=x\sqrt{x}$. Multiply the expressions: $\sqrt{x}\cdot x=x\sqrt{x}$.

Flashcard 16: What is the definition of $(f-g)(x)$ in terms of $f(x)$ and $g(x)$?

Answer: $(f-g)(x)=f(x)-g(x)$. Subtract the second function's output from the first at each input.

Flashcard 17: For $f(x)=ab^x$, which condition on $b$ gives exponential growth?

Answer: $b>1$. Base greater than 1 creates increasing exponential behavior.

Flashcard 18: Simplify $(fg)(x)$ if $f(x)=2^x$ and $g(x)=3^x$.

Answer: $(fg)(x)=6^x$. Use property $(ab)^x=a^x b^x$: $2^x\cdot 3^x=(2\cdot 3)^x=6^x$.

Flashcard 19: In $T(t)=A+Be^{-kt}$ for cooling, what does the constant $A$ represent?

Answer: Ambient (surrounding) temperature. The final temperature the object approaches as time goes to infinity.

Flashcard 20: What is $h(0)$ for $h(x)=4+2^x$?

Answer: $5$. Substitute $x=0$: $4+2^0=4+1=5$.

Flashcard 21: Find the domain of $h(x)=\frac{2^x}{x^2-9}$.

Answer: All real $x$ except $x=-3$ and $x=3$. Denominator $x^2-9=(x-3)(x+3)$ cannot equal zero.

Flashcard 22: What is the key restriction added to the domain when forming $(\frac{f}{g})(x)$?

Answer: Exclude inputs where $g(x)=0$. Division by zero is undefined, so these points are excluded.

Flashcard 23: For $f(x)=ab^x$, which condition on $b$ gives exponential growth?

Answer: $b>1$. Base greater than 1 creates increasing exponential behavior.

Flashcard 24: What is the domain of $h(x)=\frac{x+1}{x-2}$?

Answer: All real $x$ except $x=2$. Denominator cannot be zero, so exclude $x=2$.

Flashcard 25: Build $T(t)$ if ambient is $20$ and initial is $80$ with decay factor $0.8$ per hour.

Answer: $T(t)=20+60(0.8)^t$. Initial difference is $80-20=60$, so $T(t)=20+60(0.8)^t$.

Flashcard 26: Find $(f-g)(x)$ if $f(x)=x^2+4$ and $g(x)=3x$.

Answer: $(f-g)(x)=x^2-3x+4$. Subtract the expressions: $(x^2+4)-3x=x^2-3x+4$.

Flashcard 27: For $T(t)=70+20\left(\frac{1}{2}\right)^t$, what is $T(1)$?

Answer: $80$. Substitute $t=1$: $70+20\left(\frac{1}{2}\right)^1=70+20(0.5)=80$.

Flashcard 28: What is the domain of $h(x)=\frac{x+3}{x-3}$?

Answer: All real $x$ except $x=3$. Denominator cannot equal zero, so exclude $x=3$.

Flashcard 29: In $T(t)=A+Be^{-kt}$, what is the initial temperature $T(0)$ in terms of $A$ and $B$?

Answer: $T(0)=A+B$. Substitute $t=0$ into the expression and simplify.

Flashcard 30: Find $(f+g)(x)$ if $f(x)=2x-1$ and $g(x)=x^2$.

Answer: $(f+g)(x)=x^2+2x-1$. Add the expressions: $(2x-1)+x^2=x^2+2x-1$.

Flashcard 31: What is the domain of $(\frac{f}{g})(x)$ in terms of $f$, $g$, and $g(x)$?

Answer: $\text{Dom}(\frac{f}{g})=\text{Dom}(f)\cap\text{Dom}(g)\setminus\{g(x)=0\}$. Must exclude points where the denominator equals zero.

Flashcard 32: Identify the function type of $h(x)=x^2+3^x$.

Answer: Sum of a quadratic and an exponential. Combines polynomial and exponential function types through addition.

Flashcard 33: If $T(t)=A+Be^{-kt}$ with $k>0$, what is $\lim_{t\to\infty}T(t)$?

Answer: $A$. The exponential term approaches zero as $t$ approaches infinity.

Flashcard 34: What is the domain of $h(x)=\sqrt{x-3}+2^x$?

Answer: $x\ge 3$. Square root requires $x-3\ge 0$, so $x\ge 3$.

Flashcard 35: Identify the constant function added in $h(x)=4+2^x$.

Answer: $4$. The constant function is the term without a variable.

Flashcard 36: What is the domain of $h(x)=\frac{x+3}{x-3}$?

Answer: All real $x$ except $x=3$. Denominator cannot equal zero, so exclude $x=3$.

Flashcard 37: Identify the function type of $h(x)=x^2+3^x$.

Answer: Sum of a quadratic and an exponential. Combines polynomial and exponential function types through addition.

Flashcard 38: What is the domain of $h(x)=\sqrt{x-3}+2^x$?

Answer: $x\ge 3$. Square root requires $x-3\ge 0$, so $x\ge 3$.

Flashcard 39: Find $(f+g)(x)$ if $f(x)=2x-1$ and $g(x)=x^2$.

Answer: $(f+g)(x)=x^2+2x-1$. Add the expressions: $(2x-1)+x^2=x^2+2x-1$.

Flashcard 40: What is the domain of $(\frac{f}{g})(x)$ in terms of $f$, $g$, and $g(x)$?

Answer: $\text{Dom}(\frac{f}{g})=\text{Dom}(f)\cap\text{Dom}(g)\setminus\{g(x)=0\}$. Must exclude points where the denominator equals zero.

Flashcard 41: What is $h(1)$ for $h(x)=4+2^x$?

Answer: $6$. Substitute $x=1$: $4+2^1=4+2=6$.

Flashcard 42: What is the vertical shift of $f(x)$ when forming $f(x)+c$?

Answer: Shift up by $c$ units. Positive $c$ moves the graph upward by $c$ units.

Flashcard 43: In $T(t)=A+Be^{-kt}$ for cooling, what does the constant $A$ represent?

Answer: Ambient (surrounding) temperature. The final temperature the object approaches as time goes to infinity.

Flashcard 44: Simplify $(\frac{f}{g})(x)$ if $f(x)=10^x$ and $g(x)=2^x$.

Answer: $(\frac{f}{g})(x)=5^x$. Use property $\left(\frac{a}{b}\right)^x=\frac{a^x}{b^x}$: $\frac{10^x}{2^x}=\left(\frac{10}{2}\right)^x=5^x$.

Flashcard 45: What is $(\frac{f}{g})(4)$ if $f(4)=10$ and $g(4)=-5$?

Answer: $-2$. Divide the function values: $\frac{10}{-5}=-2$.

Flashcard 46: What is the definition of $(\frac{f}{g})(x)$ in terms of $f(x)$ and $g(x)$?

Answer: $(\frac{f}{g})(x)=\frac{f(x)}{g(x)}$. Divide the first function's output by the second at each input.

Flashcard 47: Identify the constant function added in $h(x)=4+2^x$.

Answer: $4$. The constant function is the term without a variable.

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

Answer: $\frac{f}{g}(x) = \frac{x+1}{x-2}$. Form the quotient of the two expressions.

Flashcard 49: Find $h(x)$ if $f(x)=x+3$ and $g(x)=x-3$ and $h(x)=\frac{f(x)}{g(x)}$.

Answer: $h(x)=\frac{x+3}{x-3}$. Form the quotient of the two functions.

Flashcard 50: Build $T(t)$ if ambient is $68$ and initial is $98$ with decay factor $0.9$ per minute.

Answer: $T(t)=68+30(0.9)^t$. Initial difference is $98-68=30$, so $T(t)=68+30(0.9)^t$.

Flashcard 51: What is the standard form of an exponential function used in Algebra $1$: growth or decay?

Answer: $f(x)=ab^x$. Standard exponential form with base $b$ and initial value $a$.

Flashcard 52: What is $h(0)$ for $h(x)=4+2^x$?

Answer: $5$. Substitute $x=0$: $4+2^0=4+1=5$.

Flashcard 53: What is the definition of $(fg)(x)$ in terms of $f(x)$ and $g(x)$?

Answer: $(fg)(x)=f(x)g(x)$. Multiply the outputs of both functions at each input value.

Flashcard 54: Identify the function type of $h(x)=5-2\left(\frac{1}{3}\right)^x$.

Answer: Constant plus decaying exponential. Rewritten as $5+(-2)\left(\frac{1}{3}\right)^x$ shows this structure.

Flashcard 55: What is the domain of $h(x)=x\sqrt{x}$?

Answer: $x\ge 0$. Square root function requires $x\ge 0$.

Flashcard 56: Identify the error: $(f+g)(x)=f(x)\cdot g(x)$. What is the correct statement?

Answer: $(f+g)(x)=f(x)+g(x)$. Addition of functions uses $+$, not multiplication ($\cdot$).

Flashcard 57: In $T(t)=A+Be^{-kt}$ for cooling, what does the parameter $k>0$ control?

Answer: Cooling rate (how fast it approaches $A$). Larger $k$ means faster cooling toward the ambient temperature.

Flashcard 58: Build $T(t)$ if ambient is $68$ and initial is $98$ with decay factor $0.9$ per minute.

Answer: $T(t)=68+30(0.9)^t$. Initial difference is $98-68=30$, so $T(t)=68+30(0.9)^t$.

Flashcard 59: Find $h(x)$ if $f(x)=2x$ and $g(x)=5^x$ and $h(x)=f(x)-g(x)$.

Answer: $h(x)=2x-5^x$. Subtract the expressions: $2x-5^x$.

Flashcard 60: Build $P(t)$ if a population starts at $500$ and increases by $3\%$ per year plus $200$ immigrants yearly.

Answer: $P(t)=500(1.03)^t+200t$. Combines exponential growth with linear immigration growth.

Flashcard 61: What is the standard form of an exponential function used in Algebra 1: growth or decay?

Answer: $f(x)=ab^x$. Standard exponential form with base $b$ and initial value $a$.

Flashcard 62: Identify the error: $(\frac{f}{g})(x)$ has domain $\text{Dom}(f)\cap\text{Dom}(g)$. What is missing?

Answer: Also require $g(x)\ne 0$. Division requires the additional restriction that $g(x)\ne 0$.

Flashcard 63: Simplify $(fg)(x)$ if $f(x)=2^x$ and $g(x)=3^x$.

Answer: $(fg)(x)=6^x$. Use property $(ab)^x=a^x b^x$: $2^x\cdot 3^x=(2\cdot 3)^x=6^x$.

Flashcard 64: Build $P(t)$ if a population starts at $500$ and increases by $3\%$ per year plus $200$ immigrants yearly.

Answer: $P(t)=500(1.03)^t+200t$. Combines exponential growth with linear immigration growth.

Flashcard 65: For $f(x)=ab^x$, which condition on $b$ gives exponential decay?

Answer: $0<b<1$. Base between 0 and 1 creates decreasing exponential behavior.

Flashcard 66: What is $(f+g)(2)$ if $f(2)=7$ and $g(2)=-3$?

Answer: $4$. Add the function values: $7+(-3)=4$.

Flashcard 67: Simplify $(f-g)(x)$ if $f(x)=5^x$ and $g(x)=2\cdot 5^x$.

Answer: $(f-g)(x)=-1\cdot 5^x$. Factor out $5^x$: $5^x-2\cdot 5^x=-1\cdot 5^x$.

Flashcard 68: For $T(t)=70+20\left(\frac{1}{2}\right)^t$, what is $\lim_{t\to\infty}T(t)$?

Answer: $70$. As $t\to\infty$, the exponential term approaches zero.

Flashcard 69: Find $(f-g)(x)$ if $f(x)=x^2+4$ and $g(x)=3x$.

Answer: $(f-g)(x)=x^2-3x+4$. Subtract the expressions: $(x^2+4)-3x=x^2-3x+4$.

Flashcard 70: Identify the error: $(f+g)(x)=f(x)\cdot g(x)$. What is the correct statement?

Answer: $(f+g)(x)=f(x)+g(x)$. Addition of functions uses $+$, not multiplication ($\cdot$).

Flashcard 71: What is $h(1)$ for $h(x)=4+2^x$?

Answer: $6$. Substitute $x=1$: $4+2^1=4+2=6$.

Flashcard 72: What is the difference between $(f+g)(x)$ and $f(x)+g(x)$?

Answer: $(f+g)(x)=f(x)+g(x)$. They are identical - both notations represent the same operation.

Flashcard 73: What is the key restriction added to the domain when forming $(\frac{f}{g})(x)$?

Answer: Exclude inputs where $g(x)=0$. Division by zero is undefined, so these points are excluded.

Flashcard 74: For $T(t)=70+20\left(\frac{1}{2}\right)^t$, what is the ambient temperature?

Answer: $70$. The constant term represents the ambient temperature.

Flashcard 75: For $T(t)=70+20\left(\frac{1}{2}\right)^t$, what is $T(1)$?

Answer: $80$. Substitute $t=1$: $70+20\left(\frac{1}{2}\right)^1=70+20(0.5)=80$.

Flashcard 76: What is $(fg)(3)$ if $f(3)=-2$ and $g(3)=9$?

Answer: $-18$. Multiply the function values: $(-2)(9)=-18$.

Flashcard 77: What is the definition of $(\frac{f}{g})(x)$ in terms of $f(x)$ and $g(x)$?

Answer: $(\frac{f}{g})(x)=\frac{f(x)}{g(x)}$. Divide the first function's output by the second at each input.

Flashcard 78: What is $(fg)(3)$ if $f(3)=-2$ and $g(3)=9$?

Answer: $-18$. Multiply the function values: $(-2)(9)=-18$.

Flashcard 79: Identify the function type of $h(x)=5-2\left(\frac{1}{3}\right)^x$.

Answer: Constant plus decaying exponential. Rewritten as $5+(-2)\left(\frac{1}{3}\right)^x$ shows this structure.

Flashcard 80: Find $T(0)$ for $T(t)=70+20\left(\frac{1}{2}\right)^t$.

Answer: $90$. Substitute $t=0$: $70+20\left(\frac{1}{2}\right)^0=70+20(1)=90$.

Flashcard 81: Identify the error: $(\frac{f}{g})(x)$ has domain $\text{Dom}(f)\cap\text{Dom}(g)$. What is missing?

Answer: Also require $g(x)\ne 0$. Division requires the additional restriction that $g(x)\ne 0$.

Flashcard 82: In $T(t)=A+Be^{-kt}$ for cooling, what does the parameter $k>0$ control?

Answer: Cooling rate (how fast it approaches $A$). Larger $k$ means faster cooling toward the ambient temperature.

Flashcard 83: What is the domain of $h(x)=\sqrt{x}+\frac{1}{x}$?

Answer: $x>0$. Both square root and fraction require $x>0$.