Combine Different Function Types - Algebra 1

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

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

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

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

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

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

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

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

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

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

6. What is the domain of $(f+g)(x)$ in terms of the domains of $f$ and $g$?

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

7. What is the domain of $h(x)=x\sqrt{x}$?

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

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

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

9. What is the result of adding a constant $c$ to a function $f(x)$?

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

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

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

11. What is $(f-g)(-1)$ if $f(-1)=5$ and $g(-1)=12$?

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

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

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

13. What is the definition of $(f-g)(x)$ in terms of $f(x)$ and $g(x)$?

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

14. For $f(x)=ab^x$, which condition on $b$ gives exponential growth?

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

15. Simplify $(fg)(x)$ if $f(x)=2^x$ and $g(x)=3^x$.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

26. What is the domain of $h(x)=\frac{x+3}{x-3}$?

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

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

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

28. What is $h(1)$ for $h(x)=4+2^x$?

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

29. What is the vertical shift of $f(x)$ when forming $f(x)+c$?

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

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

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

31. What is $(\frac{f}{g})(4)$ if $f(4)=10$ and $g(4)=-5$?

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

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

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

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

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

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

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

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

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

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

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

37. What is $h(0)$ for $h(x)=4+2^x$?

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

38. What is the definition of $(fg)(x)$ in terms of $f(x)$ and $g(x)$?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

46. For $f(x)=ab^x$, which condition on $b$ gives exponential decay?

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

47. What is $(f+g)(2)$ if $f(2)=7$ and $g(2)=-3$?

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

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

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

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

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

50. What is the difference between $(f+g)(x)$ and $f(x)+g(x)$?

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

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

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

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

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

53. What is the definition of $(\frac{f}{g})(x)$ in terms of $f(x)$ and $g(x)$?

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

54. What is $(fg)(3)$ if $f(3)=-2$ and $g(3)=9$?

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

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

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

56. What is the domain of $h(x)=\sqrt{x}+\frac{1}{x}$?

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

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