Given: f(x) = -1/(x-3) + 2. Write down the equations of the asymptotes of f, the domain of f, determine the coordinates of the x-intercept of f, write down the coordinates of the y-intercept of f, and draw the graph of f.

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Asked on August 26, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-08-26T20:03:30.898Z

5.1: Write down the equations of the asymptotes of f. Step 1: Identify the form of the function. The function is given as f(x) = (-1)/(x-3) + 2, which is in the form f(x) = (a)/(x-h) + k. Step 2: Determine the vertical and horizontal asymptotes. The vertical asymptote is x=h and the horizontal asymptote is y=k. From the given function, h=3 and k=2. The equations of the asymptotes are: x = 3 y = 2 5.2: Write down the domain of f. Step 1: Identify values for which the denominator is zero. The denominator of the function is x-3. The function is undefined when x-3=0. Step 2: Solve for x. x-3=0 x=3. The domain is all real numbers except where the function is undefined. The domain of f is x R, x ≠ 3. 5.3: Determine the coordinates of the x-intercept of f. Step 1: Set f(x)=0 to find the x-intercept. 0 = (-1)/(x-3) + 2 Step 2: Solve for x. -2 = (-1)/(x-3) 2 = (1)/(x-3) 2(x-3) = 1 2x - 6 = 1 2x = 7 x = (7)/(2) The coordinates of the x-intercept are ((7)/(2), 0). 5.4: Write down the coordinates of the y-intercept of f. Step 1: Set x=0 to find the y-intercept. f(0) = (-1)/(0-3) + 2 Step 2: Calculate f(0). f(0) = (-1)/(-3) + 2 f(0) = (1)/(3) + 2 f(0) = (1)/(3) + (6)/(3) f(0) = (7)/(3) The coordinates of the y-intercept are (0, (7)/(3)). 5.5: Draw the graph of f. Clearly show ALL the asymptotes and intercepts with the axes. To draw the graph: • Draw the vertical asymptote at x=3 as a dashed line. • Draw the horizontal asymptote at y=2 as a dashed line. • Plot the x-intercept at ((7)/(2), 0) or (3.5, 0). • Plot the y-intercept at (0, (7)/(3)) or (0, ≈ 2.33). • Since the numerator is negative (-1), the branches of the hyperbola will be in the second and fourth quadrants relative to the asymptotes. Sketch the two branches of the hyperbola approaching the asymptotes and passing through the intercepts. Send me the next one 📸

Accepted Answer · 2026-08-26T20:03:30.898Z

5.1: Write down the equations of the asymptotes of f. Step 1: Identify the form of the function. The function is given as f(x) = (-1)/(x-3) + 2, which is in the form f(x) = (a)/(x-h) + k. Step 2: Determine the vertical and horizontal asymptotes. The vertical asymptote is x=h and the horizontal asymptote is y=k. From the given function, h=3 and k=2. The equations of the asymptotes are: x = 3 y = 2 5.2: Write down the domain of f. Step 1: Identify values for which the denominator is zero. The denominator of the function is x-3. The function is undefined when x-3=0. Step 2: Solve for x. x-3=0 x=3. The domain is all real numbers except where the function is undefined. The domain of f is x R, x ≠ 3. 5.3: Determine the coordinates of the x-intercept of f. Step 1: Set f(x)=0 to find the x-intercept. 0 = (-1)/(x-3) + 2 Step 2: Solve for x. -2 = (-1)/(x-3) 2 = (1)/(x-3) 2(x-3) = 1 2x - 6 = 1 2x = 7 x = (7)/(2) The coordinates of the x-intercept are ((7)/(2), 0). 5.4: Write down the coordinates of the y-intercept of f. Step 1: Set x=0 to find the y-intercept. f(0) = (-1)/(0-3) + 2 Step 2: Calculate f(0). f(0) = (-1)/(-3) + 2 f(0) = (1)/(3) + 2 f(0) = (1)/(3) + (6)/(3) f(0) = (7)/(3) The coordinates of the y-intercept are (0, (7)/(3)). 5.5: Draw the graph of f. Clearly show ALL the asymptotes and intercepts with the axes. To draw the graph: • Draw the vertical asymptote at x=3 as a dashed line. • Draw the horizontal asymptote at y=2 as a dashed line. • Plot the x-intercept at ((7)/(2), 0) or (3.5, 0). • Plot the y-intercept at (0, (7)/(3)) or (0, ≈ 2.33). • Since the numerator is negative (-1), the branches of the hyperbola will be in the second and fourth quadrants relative to the asymptotes. Sketch the two branches of the hyperbola approaching the asymptotes and passing through the intercepts. Send me the next one 📸

Given: f(x) = -1/(x-3) + 2. Write down the equations of the asymptotes of f, the domain of f, determine the coordinates of the x-intercept of f, write down the coordinates of the y-intercept of f, and draw the graph of f.

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Given: f(x) = -1/(x-3) + 2. Write down the equations of the asymptotes of f, the domain of f, determine the coordinates of the x-intercept of f, write down the coordinates of the y-intercept of f, and draw the graph of f.

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