Solve g(x) = -x + 4: Finding Where f(x) < g(x) Using Graphs

Q: How do I know which function is above or below?

Look at the y-values on the graph! The function with the higher y-value at any given x is above the other. Between x=1 and x=4, the line g(x) is above the parabola f(x).

Q: Why don't we include the intersection points?

At intersection points, the functions are equal, not one less than the other. Since we want $ f(x) (strictly less than), we exclude x=1 and x=4 where \( f(x) = g(x) $.

Q: What if the parabola was above the line instead?

Then the answer would be the opposite intervals! You'd have $ f(x) > g(x) $ between the intersections, and \( f(x) outside them.

Q: How can I identify the intersection points from the graph?

Look for points labeled on the graph or where the curves cross each other. Here, points B(4,0) and C(1,3) show where the line and parabola intersect.

Q: Do I need to solve algebraically or can I just read the graph?

For this problem, you can read directly from the graph! The visual shows clearly where one function is below the other. Just make sure to identify the correct interval notation.

Function Comparison with Graphical Analysis

The following function is graphed below:

$g(x)=-x+4$

For which values of x is

$f(x) < g(x)$ true?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 For which values is the line greater than the parabola?

00:03 Let's compare the functions to find their intersection points

00:09 Let's arrange the equation so that one side equals 0

00:16 Let's use the quadratic formulas

00:26 These are the intersection points

00:31 Using the line, we'll find the domains where the line is greater

00:38 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The following function is graphed below:

$g(x)=-x+4$

For which values of x is

$f(x) < g(x)$ true?

Step-by-step solution

To solve this problem, we start by analyzing the graph of both functions. The line $g(x) = -x + 4$ and a parabola $f(x)$ intersect at points labeled $B$ and $C$ . We observe the behavior of these functions within the interval determined by these intersection points.

Step 1: Identify intersection points from the graph. Points $B(4,0)$ and $C(1,3)$ are where $g(x)$ is equal to $f(x)$ .
Step 2: Analyze the graph to establish where $f(x) < g(x)$ occurs. From the graph, this occurs when the parabola (representing $f(x)$ ) is below the line $g(x) = -x + 4$ .
Step 3: The region where $f(x)$ is below $g(x)$ is between the points of intersection. On the graph, this is between $x = 1$ and $x = 4$ .

Therefore, the solution is within the interval $1 < x < 4$ , during which $f(x) < g(x)$ .

Thus, the solution to the problem is $1 < x < 4$ .

Final Answer

$1 < x < 4$

Key Points to Remember

Essential concepts to master this topic

Intersection Rule: Functions are equal where their graphs meet
Technique: Compare heights between curves; $f(x) < g(x)$ when f is below g
Check: Verify endpoints: at x=1 and x=4, both functions have same value ✓

Common Mistakes

Avoid these frequent errors

Including intersection points in the solution
Don't write $1 ≤ x ≤ 4$ when finding where $f(x) < g(x)$ = wrong because at x=1 and x=4, f(x) equals g(x), not less than! The inequality is strict (<), not inclusive (≤). Always use open intervals like $1 < x < 4$ for strict inequalities.

Practice Quiz

Test your knowledge with interactive questions

The following functions are graphed below:

$$ f(x)=x^2-6x+8 $$

$$ g(x)=4x-17 $$

For which values of x is
$$ f(x)<0 $$ true?

FAQ

Everything you need to know about this question

How do I know which function is above or below?

Look at the y-values on the graph! The function with the higher y-value at any given x is above the other. Between x=1 and x=4, the line g(x) is above the parabola f(x).

Why don't we include the intersection points?

At intersection points, the functions are equal, not one less than the other. Since we want $f(x) < g(x)$ (strictly less than), we exclude x=1 and x=4 where $f(x) = g(x)$ .

What if the parabola was above the line instead?

Then the answer would be the opposite intervals! You'd have $f(x) > g(x)$ between the intersections, and $f(x) < g(x)$ outside them.

How can I identify the intersection points from the graph?

Look for points labeled on the graph or where the curves cross each other. Here, points B(4,0) and C(1,3) show where the line and parabola intersect.

Do I need to solve algebraically or can I just read the graph?

For this problem, you can read directly from the graph! The visual shows clearly where one function is below the other. Just make sure to identify the correct interval notation.

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