Question

1. f(x)= x + 1. Find the domain, range, and
intercepts of the function. Then find the
minimum and maximum values on the interval
[0, 3].

Answer 1

The domain of the function is $x \in [0,3]$

The range of the function is $f(x) \in [1, 4]$

The intercept of the function is $(x,y) = (0, 1)$

The minimum of the function is $y = 1$

The maximum of the function is $y = 4$.

Given a closed interval, the minimum and maximum values for the function are contained in its extremes. The domain of the function represents the set of values of $x$ within the interval, where the range is for the set of values of $y$ within the interval. The intercepts of the linear function have the following two forms: (i) $(x, 0)$, (ii) $(0, y)$

Domain - The domain of $f(x) = x + 1$ is $x \in [0,3]$.

Range - The range of $f(x) = x + 1$ is found below:

Lower bound

$f(0) = 0 + 1$

$f(0) = 1$

Upper bound

$f(3) = 3 + 1$

$f(3) = 4$

The range of $f(x) = x + 1$ is $f(x) \in [1, 4]$.

Intercepts - We proceed to find all intercepts:

x = 0

$f(0) = 1$

$y = 0$

$0 = x + 1$

$x = -1$

This intercept does not belong to given interval.

(0, 1) is the only intercept of  $f(x) = x + 1$.

Minimum - The minimum value of the interval is $y = 1$.

Maximum - The maximum value of the interval is $y = 4$.

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