Question

Suppose that the x-intercepts of the graph of y = f(x) are -5 and 3. . a) What are the x-intercepts of the graph of y = 7f(x)?

Answer 1

`(-5)` and
`3`.

Explanation:

The
`x`-intercepts of a graph refer to points where the graph intersects the
`x\!`-axis. These are
`x\!\!` values for which
`f(x) = 0`.

For example, since
`x = (-5)` is one of the
`x`-intercepts of
`y = f(x)`, it must be true that
`f(-5) = 0`. Likewise, since
`3` is one of the
`x`-intercepts of
`y = f(x)\!`,
`f(3) = 0`.

Since
`f(-5) = 0`, the expression
`7\, f(-5)` would also evaluate to
`0` (that is,
`7\, f(-5) = (7)\, (0) = 0`.) Thus,
`x = (-5)` would be an
`x`-intercept of the new graph
`y = 7\, f(x)`.

Likewise, since
`f(3) = 0`,
`7\, f(3) = (7)\, (0) = 0`, such that
`x = 3` would also be an
`x`-intercept of the new graph
`y = 7\, f(x)`.

No other points could be an
`x`-intercept of the new graph
`y = 7\, f(x)` without being an
`x\!`-intercept of
`y = f(x)`. For example, assume that
`x = x_(0)` is an
`x`-intercept of
`y = 7\, f(x)` but not
`y = f(x)`.
`7\, f(x_(0)) = 0`, such that
`f(x_(0)) = (1/7)\, (7\, f(x_(0))) = (1/7)\, (0) = 0`- contradiction.

Therefore, the
`x`-intercepts of the new graph would be
`x = (-5)` and
`x = 3`.