Analyze Function Statements - International Brotherhood of Electrical Workers Exam IBEW

1. What is the correct conclusion about the parity of $f(x)=x^2$?

   Even function. It satisfies $f(-x) = f(x)$ for all x in the domain.

2. Identify the $x$-intercept of $f(x)=2x-8$.

   $x=4$. Setting $f(x)=0$ and solving gives the x-value where the graph crosses the x-axis.

3. Identify the domain of $f(x)=\sqrt{x-5}$.

   $x\ge 5$. The radicand must be non-negative for the square root to be defined in real numbers.

4. Identify whether the relation $\{(0,1),(1,1),(2,1)\}$ is a function.

   Function. Each unique input corresponds to exactly one output, satisfying the function criteria.

5. Identify whether $f(x)=2x-7$ is increasing, decreasing, or constant.

   Increasing. The positive slope means outputs increase as inputs increase.

6. Identify whether the conclusion "If $f(a)=f(b)$, then $a=b$" is always true for all functions.

   Not always true. Non-injective functions allow different inputs to map to the same output.

7. What is the definition of a function in terms of inputs and outputs?

   Each input has exactly one output. A function assigns precisely one output to each input in its domain.

8. Identify whether $f(x)=-3x+1$ is increasing, decreasing, or constant.

   Decreasing. The negative slope means outputs decrease as inputs increase.

9. What is the correct conclusion about the slope of a constant function $f(x)=c$?

   Slope is $0$. The output remains unchanged regardless of input, yielding zero rate of change.

10. What conclusion is always valid about the graph of $y=f(x)$ passing the vertical line test?

    It represents a function. Passing ensures every x-value has at most one corresponding y-value.

11. Identify whether $f(x)=x^2$ is one-to-one on all real numbers.

    No; it fails the horizontal line test. Different inputs like 1 and -1 yield the same output, failing injectivity.

12. Identify the domain of $f(x)=\frac{\sqrt{x}}{x-1}$.

    $x\ge 0$ and $x\ne 1$. The radicand requires $x \ge 0$, and the denominator must not be zero.

13. What test determines whether a function is one-to-one using its graph?

    Horizontal line test. If no horizontal line intersects the graph more than once, the function is injective.

14. What is the vertical line test used to conclude about a graph?

    It determines whether the graph represents a function. The test checks if any vertical line intersects the graph more than once, indicating multiple outputs for an input.

15. Identify whether $f(x)=x^3$ is one-to-one on all real numbers.

    Yes; it passes the horizontal line test. The function is strictly increasing, ensuring unique outputs for unique inputs.

16. What is the correct conclusion about the range of $f(x)=|x|$?

    Range is $y\ge 0$. The absolute value of any real number is non-negative, covering all non-negative outputs.

17. What is the correct conclusion about the parity of $f(x)=x^3$?

    Odd function. It satisfies $f(-x) = -f(x)$ for all x in the domain.

18. What conclusion is valid if a graph fails the vertical line test?

    It is not a function. Failure means at least one x-value corresponds to multiple y-values.

19. What is the correct conclusion about the range of $f(x)=x^2$?

    Range is $y\ge 0$. Squaring any real number produces non-negative values, with all non-negative outputs achievable.

20. Identify whether $f(x)=\sqrt{x^2}$ equals $x$ for all real $x$.

    No; $\sqrt{x^2}=|x|$. The principal square root returns the non-negative value, equaling $|x|$.

21. Identify whether the relation $\{(1,2),(1,3),(2,4)\}$ is a function.

    Not a function. The input 1 maps to two different outputs, violating the function definition.

22. Identify the $y$-intercept of $f(x)=5x+3$.

    $y=3$. Evaluating $f(0)$ gives the y-value where the graph crosses the y-axis.

23. Identify whether the inverse of $f(x)=x^2$ is a function on all real numbers.

    No; the inverse relation is not a function. The function is not one-to-one, so its inverse relation maps some inputs to multiple outputs.

24. Identify the domain of $f(x)=\frac{1}{x+2}$.

    $x\ne -2$. The denominator cannot be zero to avoid division by zero.

25. Identify whether the conclusion "$f(2)=f(3)$ implies $2=3$" is valid for one-to-one functions.

    Valid. One-to-one functions require that equal outputs imply equal inputs.

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