Question

Find​ (a) the solution to the equation ​f(x)equals​0, ​(b) the​ x-intercept of the graph of yequals​f(x), and​ (c) the zero of​ f(x).
​f(x)equalsseven sixths
xminus7

What is the solution to ​f(x)equals​0?
x=

What is the​ x-intercept?

What is the zero of​ f(x)?

Answer 1

To solve the equation \( f(x) = 0 \) for the function \( f(x) = \frac{7}{6}x - 7 \), we'll follow the instructions given for parts (a), (b), and (c).

### (a) Solve \( f(x) = 0 \)

Setting the function equal to zero:

\[
\frac{7}{6}x - 7 = 0
\]

To solve for \( x \), first add \( 7 \) to both sides:

\[
\frac{7}{6}x = 7
\]

Next, multiply both sides by \( \frac{6}{7} \) to isolate \( x \):

\[
x = 7 \cdot \frac{6}{7}
\]

Simplifying gives:

\[
x = 6
\]

So, the solution to \( f(x) = 0 \) is:

\[
\boxed{6}
\]

### (b) What is the x-intercept?

The x-intercept of a graph occurs where \( y = f(x) = 0 \). Since we found that \( f(x) = 0 \) at \( x = 6 \), the x-intercept is:

\[
(6, 0)
\]

The x-intercept is:

\[
\boxed{6}
\]

### (c) What is the zero of \( f(x) \)?

The zero of a function is the value of \( x \) for which \( f(x) = 0 \). Since we just found that when \( f(x) = 0 \), \( x = 6 \), the zero of \( f(x) \) is:

\[
\boxed{6}
\]

In summary:
- \( f(x) = 0 \) solution: \( x = 6 \)
- x-intercept: \( 6 \)
- Zero of \( f(x) \): \( 6 \)