Question

the function $f(x)$ is defined as $f(x)=x+3$ the function $g(x)$ is represented on the graph below. draw (graph with grid, axes, and a line with two blue dots) compare the key features of $f(x)$ and $g(x)$ by filling i

Explanation:

Step1: Analyze \( f(x) \)

The function \( f(x) = x + 3 \) is a linear function. Its slope \( m \) is \( 1 \) (since it's in the form \( y = mx + b \), where \( m = 1 \)) and the y - intercept \( b \) is \( 3 \). To find the x - intercept, set \( y = 0 \): \( 0=x + 3\Rightarrow x=-3 \).

Step2: Analyze \( g(x) \) from the graph

• Slope calculation: Pick two points on the graph of \( g(x) \). Let's take the points \((-2,-3)\) and \((6,5)\) (from the blue dots). The slope \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{5-(-3)}{6-(-2)}=\frac{8}{8} = 1\).
• Y - intercept: The graph of \( g(x) \) crosses the y - axis at \( (0,-1) \)? Wait, no, let's re - check. Wait, when \( x = 0 \), from the line, let's see the points. Wait, the line passes through \((-2,-3)\) and \((6,5)\). Using the slope - intercept form \( y=mx + b \), with \( m = 1 \) and the point \((-2,-3)\): \( - 3=1\times(-2)+b\Rightarrow - 3=-2 + b\Rightarrow b=-1 \). Wait, but maybe I made a mistake. Wait, let's take another point. Wait, the line also passes through \((0,-1)\)? Wait, no, when \( x = 0 \), if \( y=mx + b \) and \( m = 1 \), and using the point \((6,5)\): \( 5=1\times6 + b\Rightarrow b=5 - 6=-1 \). So the equation of \( g(x) \) is \( y=x - 1 \).
• X - intercept: Set \( y = 0 \) in \( g(x)=x - 1 \), then \( 0=x - 1\Rightarrow x = 1 \).

Step3: Compare key features

• Slope: Both \( f(x) \) and \( g(x) \) have a slope of \( 1 \), so they are parallel (since parallel lines have the same slope).
• Y - intercept: \( f(x) \) has a y - intercept of \( 3 \), \( g(x) \) has a y - intercept of \( - 1 \).
• X - intercept: \( f(x) \) has an x - intercept of \( - 3 \), \( g(x) \) has an x - intercept of \( 1 \).
• Domain and Range: Both are linear functions, so their domain and range are all real numbers (\( (-\infty,\infty) \)) for both functions.

(Assuming the question is to compare key features like slope, intercepts, domain, range etc. If the question was different, for example, to find where \( f(x)=g(x) \), set \( x + 3=x - 1 \), which gives \( 3=-1 \), which is a contradiction, so they never intersect (which makes sense as they are parallel with different y - intercepts))

Answer:

• Slope: Both \( f(x) \) and \( g(x) \) have a slope of \( 1 \).
• Y - intercept: \( f(x) \) has \( y - \)intercept \( 3 \), \( g(x) \) has \( y - \)intercept \( - 1 \).
• X - intercept: \( f(x) \) has \( x - \)intercept \( - 3 \), \( g(x) \) has \( x - \)intercept \( 1 \).
• Domain/Range: Both have domain and range \( (-\infty,\infty) \).
• Intersection: Since they have the same slope and different y - intercepts, the lines are parallel and do not intersect.