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Question

Find the [tex]\( x \)[/tex]-intercepts and [tex]\( y \)[/tex]-intercept of the following function:

[tex]\[ f(x) = (x-3)(x-1)(x+3) \][/tex]

Write your answer in coordinate pairs of the form [tex]\((x, y)\)[/tex].


Sagot :

To find the [tex]$x$[/tex]-intercepts and the [tex]$y$[/tex]-intercept of the function [tex]\( f(x) = (x-3)(x-1)(x+3) \)[/tex], we'll proceed step-by-step.

### Step 1: Finding the [tex]$x$[/tex]-Intercepts
The [tex]$x$[/tex]-intercepts occur where the function [tex]\( f(x) \)[/tex] equals zero. That is, we need to solve the equation:
[tex]\[ (x-3)(x-1)(x+3) = 0 \][/tex]
For the product of these factors to equal zero, at least one of the factors must be zero. Therefore, we solve each factor individually:
[tex]\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \][/tex]
[tex]\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \][/tex]
[tex]\[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \][/tex]
Thus, the [tex]$x$[/tex]-intercepts are at the points [tex]\((3, 0)\)[/tex], [tex]\((1, 0)\)[/tex], and [tex]\((-3, 0)\)[/tex].

### Step 2: Finding the [tex]$y$[/tex]-Intercept
The [tex]$y$[/tex]-intercept occurs where [tex]\( x = 0 \)[/tex]. We need to evaluate the function [tex]\( f(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = (0-3)(0-1)(0+3) \][/tex]
Calculating inside the parentheses:
[tex]\[ f(0) = (-3)(-1)(3) \][/tex]
Multiplying these values together:
[tex]\[ f(0) = (-3) \times (-1) \times 3 = 9 \][/tex]
Therefore, the [tex]$y$[/tex]-intercept is at the point [tex]\((0, 9)\)[/tex].

### Summary
The [tex]$x$[/tex]-intercepts and [tex]$y$[/tex]-intercept of the function [tex]\( f(x) = (x-3)(x-1)(x+3) \)[/tex] are:

- [tex]$x$[/tex]-intercepts: [tex]\((-3, 0)\)[/tex], [tex]\((1, 0)\)[/tex], [tex]\((3, 0)\)[/tex]
- [tex]$y$[/tex]-intercept: [tex]\((0, 9)\)[/tex]

These are the coordinate pairs where the function intersects the respective axes.