Question

what are the domain, range, intercepts, asymptote, and the end behavior for this function?
domain:
range:
intercepts:
asymptote:
end behavior:
drag the points on the graph to change its shape. drag the gray point to change the location of the asymptote.
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Explanation:

Step1: Identify domain (all x-values)

The graph extends infinitely left and right, so all real numbers are allowed for $x$.
Domain: All real numbers, or $(-\infty, \infty)$

Step2: Identify range (all y-values)

The graph lies entirely above the x-axis ($y=0$) and approaches but never touches $y=0$, and goes upward infinitely.
Range: $y > 0$, or $(0, \infty)$

Step3: Find intercepts

• x-intercept: The graph never crosses the x-axis, so no x-intercept.
• y-intercept: The graph crosses the y-axis at $(0,1)$.

Intercepts: y-intercept at $(0,1)$; no x-intercept

Step4: Identify asymptote

The graph approaches the line $y=0$ but never touches it.
Asymptote: Horizontal asymptote $y=0$

Step5: Determine end behavior

• As $x$ approaches $-\infty$: The graph approaches $y=0$.
• As $x$ approaches $\infty$: The graph increases toward $\infty$.

End behavior: $\text{As } x \to -\infty, y \to 0; \text{As } x \to \infty, y \to \infty$

Answer:

domain: $(-\infty, \infty)$
range: $(0, \infty)$
intercepts: y-intercept at $(0, 1)$; no x-intercept
asymptote: $y=0$ (horizontal asymptote)
end behavior: As $x \to -\infty$, $y \to 0$; As $x \to \infty$, $y \to \infty$