Functions and Function Notation - Quantitative Reasoning

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

   $f(x)$ is the output value of function $f$ for input $x$. This notation evaluates the function (f) at the specific input value (x), yielding the corresponding output.

2. What is the domain of $f(x)=\sqrt{x+1}$ in interval notation?

   $[-1,\infty)$. Require the expression inside the square root to be non-negative to yield real outputs.

3. What is the domain of $f(x)=\sqrt{5-x}$ in interval notation?

   $(-\infty,5]$. Ensure the radicand is non-negative for the square root function to be defined over reals.

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

   Input $a$ maps to output $b$ under $f$. The equation states that applying function (f) to input (a) produces the output value (b).

5. What is $g(f(x))$ if $f(x)=x^2$ and $g(x)=x-1$?

   $g(f(x))=x^2-1$. Input (f(x)) into (g), resulting in subtraction after squaring.

6. What is $(g\circ f)(2)$ if $f(x)=x+3$ and $g(x)=2x$?

   $(g\circ f)(2)=10$. Evaluate (f(2)) first, then apply (g) to that output for the composition.

7. What is the key distinction between $f^{-1}(x)$ and $\frac{1}{f(x)}$?

   $f^{-1}(x)$ is inverse; $\frac{1}{f(x)}$ is reciprocal. The inverse undoes the function's operation, while the reciprocal is the multiplicative inverse of the output.

8. What is the range of a function (in words) in function notation context?

   The set of all possible output values $f(x)$. The range includes every possible output obtained by applying the function to values in its domain.

9. What is the value of $f(1)$ if $f(x)=\begin{cases}x+1,&x<2\\3x,&x\ge 2\end{cases}$?

   $f(1)=2$. As (1 < 2), apply the first piece of the piecewise definition for computation.

10. What is the meaning of $f(x)=f(y)$ for a one-to-one function $f$?

    It implies $x=y$. One-to-one functions are injective, so equal outputs necessitate equal inputs.

11. What is the value of $f(3)-f(1)$ if $f(x)=x^2+2$?

    $f(3)-f(1)=8$. Compute each function value separately using the quadratic formula, then subtract.

12. What is the value of $f(2)$ if $f(x)=\begin{cases}x+1,&x<2\\3x,&x\ge 2\end{cases}$?

    $f(2)=6$. Since (2 geq 2), use the second piece of the piecewise function to evaluate.

13. What is $(f\circ g)(2)$ if $f(x)=x+3$ and $g(x)=2x$?

    $(f\circ g)(2)=7$. First compute (g(2)), then input that result into (f) to find the composed value.

14. What is the definition of an inverse function in function notation?

    $f^{-1}(y)=x$ exactly when $f(x)=y$. The inverse function reverses the mapping, so inputs and outputs are swapped.

15. What is $f(x-3)$ if $f(x)=x^2$?

    $f(x-3)=(x-3)^2$. Substitute the expression (x-3) into the quadratic function in place of (x).

16. What is the composition notation $(f\circ g)(x)$ equal to?

    $(f\circ g)(x)=f(g(x))$. Function composition applies (f) to the result of (g(x)), nesting the functions.

17. What is the domain of a function (in words) in function notation context?

    The set of all allowed input values $x$. The domain consists of every input (x) for which the function is defined and produces a valid output.

18. What is the domain of $f(x)=\frac{1}{x-5}$ in interval notation?

    $(-\infty,5)\cup(5,\infty)$. Exclude the value where the denominator is zero to ensure the function is defined for all real inputs.

19. What is $f(g(x))$ if $f(x)=x^2$ and $g(x)=x-1$?

    $f(g(x))=(x-1)^2$. Substitute (g(x)) into (f) as its argument, yielding the squared expression.

20. What is the value of $f(-2)$ if $f(x)=3x-5$?

    $f(-2)=-11$. Substitute (x = -2) into the linear function to compute the output directly.

21. What is $f^{-1}(9)$ if $f(x)=x^2$ with domain restricted to $x\ge 0$?

    $f^{-1}(9)=3$. With the domain restriction ensuring one-to-one, solve ($x^2$ = 9) and take the non-negative root.

22. What is the value of $g(4)$ if $g(x)=x^2-1$?

    $g(4)=15$. Evaluate the quadratic function by plugging in (x = 4) and simplifying.

23. What is the value of $f(a+1)$ if $f(x)=2x-7$?

    $f(a+1)=2a-5$. Replace (x) with (a+1) in the linear expression and simplify algebraically.

24. What is the value of $h(0)$ if $h(x)=\frac{x+2}{3}$?

    $h(0)=\frac{2}{3}$. Insert (x = 0) into the rational function and reduce the fraction.

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