To determine whether each function is
one-to-one,
onto, both, or neither, we need to analyze the graphs provided. Let's go through each problem step by step.
---
Problem 7:
#### Graph:
The graph shows a horizontal line at \( y = 0 \).
#### Analysis:
1.
One-to-One: A function is one-to-one if every horizontal line intersects the graph at most once. In this case, the horizontal line \( y = 0 \) intersects the graph infinitely many times (every point on the line). Therefore, the function is
not one-to-one.
2.
Onto: A function is onto if every \( y \)-value in the codomain is mapped to by some \( x \)-value in the domain. Here, the function only maps to \( y = 0 \), so it does not cover all possible \( y \)-values. Therefore, the function is
not onto.
#### Conclusion:
The function is
neither one-to-one nor onto.
Answer: D. Neither
---
Problem 8:
#### Graph:
The graph shows a parabola opening upwards with its vertex at the origin \((0, 0)\).
#### Analysis:
1.
One-to-One: A function is one-to-one if every horizontal line intersects the graph at most once. In this case, horizontal lines above the vertex intersect the graph at two points (e.g., \( y = 1 \) intersects at two points). Therefore, the function is
not one-to-one.
2.
Onto: A function is onto if every \( y \)-value in the codomain is mapped to by some \( x \)-value in the domain. The parabola covers all \( y \)-values greater than or equal to 0 but does not cover negative \( y \)-values. Therefore, the function is
not onto.
#### Conclusion:
The function is
neither one-to-one nor onto.
Answer: D. Neither
---
Problem 9:
#### Graph:
The graph shows a curve that appears to be a hyperbola-like shape, extending from \((0, -\infty)\) to \((0, \infty)\) and approaching the \( y \)-axis as an asymptote.
#### Analysis:
1.
One-to-One: A function is one-to-one if every horizontal line intersects the graph at most once. In this case, every horizontal line intersects the graph exactly once. Therefore, the function is
one-to-one.
2.
Onto: A function is onto if every \( y \)-value in the codomain is mapped to by some \( x \)-value in the domain. The graph covers all \( y \)-values from \(-\infty\) to \(\infty\). Therefore, the function is
onto.
#### Conclusion:
The function is
both one-to-one and onto.
Answer: C. Both
---
Final Answers:
1. Problem 7:
D. Neither
2. Problem 8:
D. Neither
3. Problem 9:
C. Both
Boxed Answer:
\[
\boxed{D, D, C}
\]
Parent Tip: Review the logic above to help your child master the concept of one to one function worksheet.
