Question

Phs of two functions, f(x) and g(x), are giv f(x)=|x-5|+6 g(x)=(1)/(3)x+(17)/(3)

Answer 1

The point of intersection of the functions
`\(f(x)\)` and
`\(g(x)\)` occurs at
`\(x = 4\),` and their respective values at this point are
`\(f(4) = 7\)` and
`\(g(4) = 7\).`

Step-by-step explanation:

To find the point of intersection, set
`\(f(x)\)` equal to
`\(g(x)\)` and solve for
`\(x\):`

`\[|x - 5| + 6 = (1)/(3)x + (17)/(3).\]`

For
`\(x < 5\)`, the equation becomes:

`\[-(x - 5) + 6 = (1)/(3)x + (17)/(3),\]`

which simplifies to
`\(x = 4\).`

For
`\(x \geq 5\)`, the equation becomes:

`\[(x - 5) + 6 = (1)/(3)x + (17)/(3),\]`

which also simplifies to
`\(x = 4\)`.

Evaluate
`\(f(4)\)` and
`\(g(4)\)` to find the y-coordinates:

`\[f(4) = |4 - 5| + 6 = 7,\]`

`\[g(4) = (1)/(3)(4) + (17)/(3) = 7.\]`

Therefore, the functions intersect at the point
`\((4, 7)\),` indicating that both functions yield the same y-value at this x-value.