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Difficulty: HardLinear Functions and Graphs

A linear function ff has a slope of 23\frac{2}{3}. The function gg is defined by g(x)=f(x+4)−3g(x) = f(x + 4) - 3. If the graph of y=g(x)y = g(x) in the xyxy-plane has an xx-intercept at (5,0)(5, 0), what is the xx-intercept of the graph of y=f(x)y = f(x)?

Answer: 4.5

Answer

4.5
The correct answer is 4.54.5 (or 92\frac{9}{2}). By using the function transformation relation, we find g(5)=f(9)−3=0g(5) = f(9) - 3 = 0, which yields f(9)=3f(9) = 3. With a slope of 23\frac{2}{3}, the equation of the line is f(x)=23x−3f(x) = \frac{2}{3}x - 3. Setting f(x)=0f(x) = 0 yields the xx-intercept at x=4.5x = 4.5.

Step-by-Step Solution

1
Use the definition of function gg and its given xx-intercept to find a point on the graph of ff.
f(9)=3f(9) = 3
Since the graph of y=g(x)y = g(x) has an xx-intercept at (5,0)(5, 0), we know g(5)=0g(5) = 0. Substituting x=5x = 5 into the definition g(x)=f(x+4)−3g(x) = f(x + 4) - 3 gives g(5)=f(5+4)−3=f(9)−3g(5) = f(5 + 4) - 3 = f(9) - 3. Since g(5)=0g(5) = 0, it follows that f(9)−3=0f(9) - 3 = 0, or f(9)=3f(9) = 3.
2
Determine the equation of the linear function ff using its slope and the point identified in Step 1.
f(x)=23x−3f(x) = \frac{2}{3}x - 3
The function ff is linear with a slope of 23\frac{2}{3} and passes through the point (9,3)(9, 3). Using the point-slope formula, we get f(x)−3=23(x−9)f(x) - 3 = \frac{2}{3}(x - 9), which simplifies to f(x)=23x−3f(x) = \frac{2}{3}x - 3.
3
Find the xx-intercept of the graph of y=f(x)y = f(x) by setting f(x)=0f(x) = 0.
x=4.5x = 4.5 (or 92\frac{9}{2})
To find the xx-intercept, set f(x)=0f(x) = 0. This gives 23x−3=0\frac{2}{3}x - 3 = 0. Adding 33 to both sides and multiplying by 32\frac{3}{2} yields x=92x = \frac{9}{2}, which is equal to 4.54.5.

Key Concept

Linear Functions and Graphs

Alternative Method

Alternatively, we can write the equation of ff in slope-intercept form as f(x)=23x+bf(x) = \frac{2}{3}x + b. Then the definition of g(x)g(x) becomes g(x)=23(x+4)+b−3=23x+83+b−3=23x+b−13g(x) = \frac{2}{3}(x + 4) + b - 3 = \frac{2}{3}x + \frac{8}{3} + b - 3 = \frac{2}{3}x + b - \frac{1}{3}. Since the graph of gg has an xx-intercept at (5,0)(5, 0), we substitute x=5x = 5 and g(5)=0g(5) = 0 to get 23(5)+b−13=0\frac{2}{3}(5) + b - \frac{1}{3} = 0, which simplifies to 3+b=03 + b = 0, so b=−3b = -3. This gives f(x)=23x−3f(x) = \frac{2}{3}x - 3. Finally, the xx-intercept is found by solving 23x−3=0\frac{2}{3}x - 3 = 0, resulting in x=4.5x = 4.5.
Estimated Time:2m 0s
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Topics

Linear Functions and Graphs

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