Questions: What is the x-intercept of the following linear function? A. (0,4) f(x)=-4x+5 B. (4,0) C. (0, 5/4) D. (5/4, 0)

What is the x-intercept of the following linear function?
A. (0,4)
f(x)=-4x+5
B. (4,0)
C. (0, 5/4)
D. (5/4, 0)
Transcript text: What is the $x$-intercept of the following linear function? A. $(0,4)$ \[ f(x)=-4 x+5 \] B. $(4,0)$ C. $\left(0, \frac{5}{4}\right)$ D. $\left(\frac{5}{4}, 0\right)$
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Solution

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Solution Steps

To find the \(x\)-intercept of a linear function, set \(f(x) = 0\) and solve for \(x\). For the given function \(f(x) = -4x + 5\), set \(-4x + 5 = 0\) and solve for \(x\).

Solution Approach
  1. Set the function equal to zero: \(-4x + 5 = 0\).
  2. Solve for \(x\) to find the \(x\)-intercept.
Step 1: Determine the $x$-intercept of the linear function

The $x$-intercept of a function is the point where the graph of the function crosses the $x$-axis. At this point, the value of $y$ (or $f(x)$) is zero. Given the function:

\[ f(x) = -4x + 5 \]

To find the $x$-intercept, set \( f(x) = 0 \) and solve for \( x \):

\[ 0 = -4x + 5 \]

Add \( 4x \) to both sides:

\[ 4x = 5 \]

Divide both sides by 4:

\[ x = \frac{5}{4} \]

Thus, the $x$-intercept is \(\left(\frac{5}{4}, 0\right)\).

Final Answer for Question 7

The answer is \(\boxed{D}\).

Step 2: Identify the $x$ and $y$ intercepts from the table

The table provides several points \((x, y)\). The $x$-intercept occurs where \( y = 0 \), and the $y$-intercept occurs where \( x = 0 \).

From the table:

  • The $x$-intercept is at \((-3, 0)\).
  • The $y$-intercept is at \((0, 2)\).
Step 3: Calculate the difference between the $x$ and $y$ intercepts

The difference between the $x$-intercept and the $y$-intercept is calculated by finding the distance between the points \((-3, 0)\) and \((0, 2)\).

The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substitute the coordinates of the intercepts:

\[ d = \sqrt{(0 - (-3))^2 + (2 - 0)^2} \]

\[ d = \sqrt{3^2 + 2^2} \]

\[ d = \sqrt{9 + 4} \]

\[ d = \sqrt{13} \]

Final Answer for Question 8

The difference between the $x$ and $y$ intercepts is \(\boxed{\sqrt{13}}\).

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