Function Notation and Evaluation - Algebra 1

1. What does it mean if an input $x$ is not in the domain of $f$?

   $f(x)$ is undefined for that input. The function cannot be evaluated at that input.

2. Identify whether $f(2)$ is defined if $f(x)=\frac{1}{x-2}$.

   $f(2)$ is undefined. Division by zero occurs when $x=2$.

3. What is the domain of a function in words?

   The set of all allowable input values. Only these values can be substituted into the function.

4. What is the domain of $f(x)=\sqrt{x-5}$ in inequality form?

   $x\ge 5$. Square root requires non-negative radicand.

5. Find the domain of $f(x)=\frac{2}{x}$ in restriction form.

   All real $x$ such that $x\neq 0$. Denominator cannot equal zero.

6. Interpret $P(5)=1200$ if $P(t)$ is population after $t$ years.

   After $5$ years, the population is $1200$. Function output shows population at given time.

7. Find $f(-2)$ if $f(x)=-x+6$.

   $f(-2)=8$. Substitute $x=-2$: $f(-2)=-(-2)+6=8$.

8. What does the statement $f(x)=0$ ask you to find?

   All inputs $x$ that make the output equal to $0$. Find the zeros or roots of the function.

9. What is the difference between $f(x)$ and $f\,x$ in algebra?

   $f(x)$ is a function value; $f\,x$ is multiplication. Parentheses indicate function evaluation, not multiplication.

10. Find $q(-2)$ if $q(x)=x^2+4x+1$.

    $q(-2)=-3$. Substitute $x=-2$: $q(-2)=(-2)^2+4(-2)+1=-3$.

11. Find $s(3)$ if $s(x)=\frac{x+1}{2}$.

    $s(3)=2$. Substitute $x=3$: $s(3)=\frac{3+1}{2}=2$.

12. Find $g(x+2)$ if $g(x)=3x-1$.

    $g(x+2)=3x+5$. Replace $x$ with $(x+2)$: $g(x+2)=3(x+2)-1=3x+5$.

13. What does the statement $f(x)<0$ describe?

    Inputs $x$ where the function output is negative. The function is below the $x$-axis at these inputs.

14. Find $f(-3)$ if $f(x)=x^2-1$.

    $f(-3)=8$. Substitute $x=-3$: $f(-3)=(-3)^2-1=8$.

15. Find $p(1)$ if $p(x)=3x^2+2x$.

    $p(1)=5$. Substitute $x=1$: $p(1)=3(1)^2+2(1)=5$.

16. Find $r(5)$ if $r(x)=10-2x$.

    $r(5)=0$. Substitute $x=5$: $r(5)=10-2(5)=0$.

17. What does the statement $f(x)>0$ describe?

    Inputs $x$ where the function output is positive. The function is above the $x$-axis at these inputs.

18. Find $h(4)$ if $h(x)=2(x-1)$.

    $h(4)=6$. Substitute $x=4$: $h(4)=2(4-1)=6$.

19. Interpret $C(10)=25$ if $C(d)$ is the cost in dollars for $d$ miles.

    A $10$-mile trip costs $25$ dollars. Function notation describes real-world relationships.

20. What does $f(x)-f(a)$ represent?

    The difference between outputs at inputs $x$ and $a$. Shows the change in function values between inputs.

21. What is the difference between $f(x)$ and $f\cdot x$?

    $f(x)$ is a function value; $f\cdot x$ is multiplication. Parentheses indicate function evaluation, not multiplication.

22. What is the range of a function in words?

    The set of all possible output values. All values the function can produce as outputs.

23. Find $t(-1)$ if $t(x)=\lvert x\rvert+2$.

    $t(-1)=3$. Substitute $x=-1$: $t(-1)=\lvert-1\rvert+2=3$.

24. Find $f(2)$ if the table gives $f(2)=9$.

    $f(2)=9$. Read function value directly from table.

25. Find the domain of $f(x)=\sqrt{x+4}$ in inequality form.

    $x\ge -4$. Square root requires $x+4\ge 0$.

26. Find $h(2x)$ if $h(x)=x^2-4$.

    $h(2x)=4x^2-4$. Replace $x$ with $2x$: $h(2x)=(2x)^2-4=4x^2-4$.

27. Find $q(1)$ if $q(x)=\frac{4}{x+1}$.

    $q(1)=2$. Substitute $x=1$: $q(1)=\frac{4}{1+1}=2$.

28. Find $g(2)$ if $g(x)=\frac{1}{2}x-4$.

    $g(2)=-3$. Substitute $x=2$: $g(2)=\frac{1}{2}(2)-4=-3$.

29. What is the domain of $f(x)=\frac{1}{x+3}$ in set notation?

    All real $x$ such that $x\neq -3$. Denominator cannot equal zero, so exclude $x=-3$.

30. What is the meaning of the statement $f(3)=7$?

    When $x=3$, the function output is $7$. Substitute $x=3$ into the function to get output $7$.

31. What does $f(0)$ represent on the graph of $y=f(x)$?

    The $y$-intercept value. When $x=0$, output gives $y$-intercept.

32. Identify whether $f(-1)$ is defined if $f(x)=\sqrt{x+1}$.

    $f(-1)$ is defined and equals $0$. Square root of $0$ is defined: $f(-1)=\sqrt{0}=0$.

33. Identify the input and output in the ordered pair $(x,f(x))$.

    Input is $x$; output is $f(x)$. First coordinate is input, second is output.

34. Find $u(-4)$ if $u(x)=\lvert x+1\rvert$.

    $u(-4)=3$. Substitute $x=-4$: $u(-4)=\lvert-4+1\rvert=3$.

35. What is the domain of $f(x)=\sqrt{2-x}$ in inequality form?

    $x\le 2$. Square root requires non-negative radicand.

36. Find $h(0)$ if $h(x)=5-3x$.

    $h(0)=5$. Substitute $x=0$: $h(0)=5-3(0)=5$.

37. What is the domain of $f(x)=\frac{5}{x^2-9}$ in restriction form?

    All real $x$ such that $x\neq 3$ and $x\neq -3$. Factor: $x^2-9=(x-3)(x+3)$, exclude both zeros.

38. What point corresponds to $f(a)=b$ on the graph of $y=f(x)$?

    The point $(a,b)$. Function notation corresponds to coordinate pairs.

39. What is the meaning of the statement $f(a)=b$?

    When $x=a$, the function output is $b$. General form showing input $a$ produces output $b$.

40. Find $f(4)$ if $f(x)=2x+5$.

    $f(4)=13$. Substitute $x=4$: $f(4)=2(4)+5=13$.

41. Find the domain of $f(x)=\sqrt{x}$ in inequality form.

    $x\ge 0$. Square root requires non-negative radicand.

42. Find $p(x-1)$ if $p(x)=x^2$.

    $p(x-1)=(x-1)^2$. Replace $x$ with $(x-1)$: $p(x-1)=(x-1)^2$.

43. Find $g(6)$ if $g(x)=\frac{x}{3}+1$.

    $g(6)=3$. Substitute $x=6$: $g(6)=\frac{6}{3}+1=3$.

44. Find the domain of $f(x)=\frac{x+1}{x-7}$ in restriction form.

    All real $x$ such that $x\neq 7$. Denominator cannot equal zero.

45. What does $f(x+h)$ mean in function notation?

    The output when the input is $x+h$. Substitute the entire expression $(x+h)$ for $x$.

46. Interpret $T(3)=68$ if $T(h)$ is temperature after $h$ hours.

    After $3$ hours, the temperature is $68$ degrees. Function value represents temperature at specific time.

47. Find $p(0)$ if $p(x)=x^3-2x$.

    $p(0)=0$. Substitute $x=0$: $p(0)=0^3-2(0)=0$.

48. Find the domain of $f(x)=\frac{1}{(x-1)(x+2)}$ in restriction form.

    All real $x$ such that $x\neq 1$ and $x\neq -2$. Both factors in denominator cannot equal zero.

49. What is the standard notation for the output variable of $f(x)$?

    $y=f(x)$. Standard way to represent function output.

50. Interpret $d(2.5)=150$ if $d(t)$ is distance in miles after $t$ hours.

    After $2.5$ hours, the distance traveled is $150$ miles. Function describes distance as function of time.

51. What does the notation $f(x)$ represent in function notation?

    $f(x)$ is the output value of function $f$ for input $x$. Function notation shows the input-output relationship.

52. Find $f(a)$ if $f(x)=x^2+3$.

    $f(a)=a^2+3$. Replace $x$ with $a$ in the function expression.

53. Identify $x$ if $f(x)=12$ and the table shows $f(4)=12$.

    $x=4$. Find input that produces given output.

54. Find $m(3)$ if $m(x)=\sqrt{x+1}$.

    $m(3)=2$. Substitute $x=3$: $m(3)=\sqrt{3+1}=2$.

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