Question

7 use the picture of the graphs of g(x) and f(x) to answer the questions. remember if the graph hits the edge of the window it means that it continues on. what is the domain of g(x)? use interval notation. what is the range of g(x)? use interval notation. what is the domain of f(x)? use interval notation. g(4)= g(x)=1 has f(5)= f(x)=3 when x= the slope of f(x) is the y - intercepts of f(x) is therefore, the equation of f(x) is for what values of x does f(x)=g(x)? x= and x= when is f(x)≥g(x)? f(10)=

Explanation:

Step1: Find domain of \(g(x)\)

The graph of \(g(x)\) extends infinitely in both directions horizontally. So the domain is \((-\infty,\infty)\).

Step2: Find range of \(g(x)\)

The graph of \(g(x)\) extends infinitely in both directions vertically. So the range is \((-\infty,\infty)\).

Step3: Find domain of \(f(x)\)

The graph of \(f(x)\) extends infinitely in both directions horizontally. So the domain is \((-\infty,\infty)\).

Step4: Find \(g(4)\)

Locate \(x = 4\) on the \(x -\)axis, then go up to the graph of \(g(x)\). We see that \(g(4)=- 2\).

Step5: Solve \(g(x)=1\)

Find the points where the horizontal line \(y = 1\) intersects \(g(x)\). There are 2 intersection points, so \(g(x)=1\) has 2 solutions.

Step6: Find \(f(5)\)

Locate \(x = 5\) on the \(x -\)axis, then go up to the graph of \(f(x)\). We find \(f(5)=3\).

Step7: Solve \(f(x)=3\)

Find the \(x -\)values where \(y = 3\) on \(f(x)\). We see \(x = 5\).

Step8: Find slope of \(f(x)\)

The line \(f(x)\) passes through \((0,-2)\) and \((2,0)\). Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{0+2}{2 - 0}=1\).

Step9: Find y - intercept of \(f(x)\)

The line \(f(x)\) crosses the \(y -\)axis at \((0,-2)\), so the y - intercept is \(-2\).

Step10: Find equation of \(f(x)\)

Using the slope - intercept form \(y=mx + b\) with \(m = 1\) and \(b=-2\), the equation is \(y=x - 2\).

Step11: Find intersection points of \(f(x)\) and \(g(x)\)

The graphs of \(f(x)\) and \(g(x)\) intersect at \(x = 2\) and \(x=6\).

Step12: Find when \(f(x)\geq g(x)\)

The graph of \(f(x)\) is above or on the graph of \(g(x)\) when \(2\leq x\leq6\), so the interval is \([2,6]\).

Step13: Find \(f(10)\)

Using the equation \(f(x)=x - 2\), when \(x = 10\), \(f(10)=10 - 2=8\).

Answer:

Domain of \(g(x)\): \((-\infty,\infty)\)
Range of \(g(x)\): \((-\infty,\infty)\)
Domain of \(f(x)\): \((-\infty,\infty)\)
\(g(4)\): \(-2\)
\(g(x)=1\) has: 2 solutions
\(f(5)\): \(3\)
\(f(x)=3\) when \(x\): \(5\)
The slope of \(f(x)\): \(1\)
The y - intercepts of \(f(x)\): \(-2\)
The equation of \(f(x)\): \(y=x - 2\)
\(x\) values for \(f(x)=g(x)\): \(2\) and \(6\)
When \(f(x)\geq g(x)\): \([2,6]\)
\(f(10)\): \(8\)