Question

the graph is represents function f. which statements are true?
$f(t) = 60$ when $t = 30$ and $f(150) = 0$
$f(10) = 15$ and $f(110) > f(130)$
$f(20) = 30$ and $f(100) = 60$
$f(50) = f(90)$ and $f(120) < f(140)$

Explanation:

Step1: Analyze the graph's shape

The graph is a trapezoid - like shape with an increasing segment, a constant segment, and a decreasing segment. Let's assume the x - axis is \(t\) and y - axis is \(f(t)\).

Step2: Check each statement

• First statement: \(f(t) = 60\) when \(t = 30\) and \(f(150)=0\)
• From the graph, if we assume the horizontal part (constant) starts around \(t = 30\) with \(f(t)=60\) and the line goes down to \(f(150) = 0\) (since the end of the graph is at a low point, maybe \(t = 150\) with \(f(t)=0\)), this seems plausible.
• Second statement: \(f(10)=15\) and \(f(110)>f(130)\)
• For the increasing part, if at \(t = 10\), the slope of the increasing line: Let's say from \(t = 0\) (assuming the start point) to \(t = 30\) (where it becomes constant) with \(f(t)\) going from 0 to 60. The slope \(m=\frac{60 - 0}{30-0}=2\). So at \(t = 10\), \(f(10)=2\times10 = 20

eq15\), so \(f(10)=15\) is false. Also, for the decreasing part, as \(t\) increases from 110 to 130, \(f(t)\) is decreasing (since the line is going down), so \(f(110)>f(130)\) is true, but since the first part of the statement is false, the whole statement is false.

• Third statement: \(f(20)=30\) and \(f(100)=60\)
• Using the slope of the increasing line (slope \(m = 2\) as above), at \(t = 20\), \(f(20)=2\times20=40

eq30\), so this is false.

• Fourth statement: \(f(50)=f(90)\) and \(f(120)<f(140)\)
• For the constant part, from \(t = 30\) to \(t = 90\) (assuming the constant part is from \(t = 30\) to \(t = 90\)), \(f(t)\) is constant (60), so \(f(50)=f(90) = 60\). For the decreasing part, as \(t\) increases from 120 to 140, \(f(t)\) is decreasing, so \(f(120)>f(140)\) (since as \(t\) increases, \(f(t)\) decreases), so \(f(120)<f(140)\) is false.

Answer:

The true statement is: \(f(t)=60\) when \(t = 30\) and \(f(150)=0\) (the first option) and \(f(10)=15\) and \(f(110)>f(130)\) is partially true but the first part is false, \(f(20)=30\) and \(f(100)=60\) is false, \(f(50)=f(90)\) and \(f(120)<f(140)\) is false. So the true statement is the first one: \(f(t)=60\) when \(t = 30\) and \(f(150)=0\)