Question: Use the graph to determine a. the function's domain; b. the function's range; c. the x-intercepts, if any; d. the y-intercept, if any; and e. the missing question value, indicated by the question mark, below.
f(1) = ?
Graph approaches but never touches the x-axis.
y = f(x)
a. The domain is □. (Type your answer in interval notation. Simplify your answer.)
b. The range is □. (Type your answer in interval notation. Simplify your answer.)
c. Select the correct choice below and fill in any answer boxes within your choice.
A. The x-intercept(s) is (are) □.
(Type an integer. Use a comma to separate answers as needed.)
B. There is no x-intercept.
d. Select the correct choice below and fill in any answer boxes within your choice.
A. The y-intercept is 3. (Type an integer.)
B. There is no y-intercept.
e. f(1) = 6
Graph shape: a blue increasing curve in the top-right graph, with left side approaching the x-axis from above and never touching it, passing through the y-axis at 3 and rising steeply to about (1, 6); position_hint = top-right
1. **State the problem:** We are given a graph of a function $y = f(x)$ that approaches but never touches the x-axis, passes through the y-axis at 3, and rises steeply to the point $(1,6)$. We need to find:
a. The domain of $f$
b. The range of $f$
c. The x-intercepts, if any
d. The y-intercept, if any
e. The value $f(1)$
2. **Domain:** The graph is shown in the top-right quadrant and the curve is continuous and increasing. Since the graph extends indefinitely to the right and the left side approaches the x-axis but never touches it, the domain includes all real numbers greater than or equal to 0 (assuming the graph starts at $x=0$). If the graph extends infinitely to the right and starts at $x=0$, the domain is:
$$\text{Domain} = [0, \infty)$$
3. **Range:** The graph approaches the x-axis from above but never touches it, so the minimum value of $f(x)$ is just above 0. The graph passes through $y=3$ at $x=0$ and rises steeply to $y=6$ at $x=1$, and presumably continues increasing. Therefore, the range is all values greater than 0:
$$\text{Range} = (0, \infty)$$
4. **X-intercepts:** Since the graph approaches but never touches the x-axis, there are no x-intercepts.
$$\text{No x-intercepts}$$
5. **Y-intercept:** The graph passes through the y-axis at $y=3$, so the y-intercept is 3.
$$\text{Y-intercept} = 3$$
6. **Value of $f(1)$:** The graph passes through the point $(1,6)$, so:
$$f(1) = 6$$
**Final answers:**
- a. Domain: $[0, \infty)$
- b. Range: $(0, \infty)$
- c. X-intercepts: None
- d. Y-intercept: 3
- e. $f(1) = 6$