Acceso
Prima
Hogar Banco de preguntas Matemáticas Álgebra
Pregunta
upstudy study bank question image url

39. Write a linear function \( f \) with values \( f(-3)=5 \) and \( f(6)=11 \).

Ask by Page Turner. in the United States
Mar 11,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

Responder

The linear function is \( f(x) = \frac{2}{3}x + 7 \).

Solución

We start with the general form of a linear function: \[ f(x)=ax+b. \] Given that \( f(-3)=5 \), we have: \[ a(-3)+b=5 \quad \Rightarrow \quad -3a+b=5. \] Given that \( f(6)=11 \), we have: \[ a(6)+b=11 \quad \Rightarrow \quad 6a+b=11. \] Subtract the first equation from the second: \[ (6a+b) - (-3a+b)=11-5 \quad \Rightarrow \quad 9a=6. \] Solve for \( a \): \[ a=\frac{6}{9}=\frac{2}{3}. \] Substitute \( a=\frac{2}{3} \) into the equation \(-3a+b=5\): \[ -3\left(\frac{2}{3}\right)+b=5 \quad \Rightarrow \quad -2+b=5. \] Solve for \( b \): \[ b=5+2=7. \] Thus, the linear function is: \[ f(x)=\frac{2}{3}x+7. \]

Respondido por UpStudy AI y revisado por un tutor profesional

error msg
Explicar
Simplifique esta solución

Mind Expander

To find a linear function \( f(x) \), you'll want to determine the slope \( m \) first. You can calculate it using the points given: \( (-3, 5) \) and \( (6, 11) \). The slope \( m \) is calculated as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{11 - 5}{6 - (-3)} = \frac{6}{9} = \frac{2}{3}. \] Now, you can use the point-slope form of the equation \( y - y_1 = m(x - x_1) \) with either point. Let's use \( (-3, 5) \): \[ f(x) - 5 = \frac{2}{3}(x + 3). \] Simplifying this gives: \[ f(x) = \frac{2}{3}x + 7. \] And there you have it, \( f(x) = \frac{2}{3}x + 7 \) is your linear function! Now, if you're feeling like a mathematician, you can easily test the function's accuracy with the original values. Plug in \( -3 \) and \( 6 \) and confirm that you retrieve \( 5 \) and \( 11 \) respectively. It's like giving your math a little friendly check-up!

preguntas relacionadas

Guia \( 20 \quad 03 \quad 25 \) 1 factoriza 10s. Sig. trinomios Cuadros Perfectos 1) \( x^{2}-8 x+16 \) \[ x^{2}-8 x+16=(x-4)^{2} \] 2) \[ \begin{array}{l} x^{2}-20 x+100 \\ x^{2}-20 x+100=(x-10)^{2} \end{array} \] 3) \[ \begin{array}{l} x^{2}-24 x+144 \\ -2.12=-24 \\ x^{2}-24 x+144=(x-12)^{2} \end{array} \] 4) \( 25 x^{2}-18 x+81 \)
Álgebra Mexico Mar 21, 2025
\[ \text{¿Cuál es } 2011^2 - 2010^2 \text{?} \]
Álgebra Mexico Mar 21, 2025
Which description best explains the domain of \( (g \circ f)(x) \) ? the elements in the domain of \( f(x) \) for which \( g(f(x)) \) is defined the elements in the domain of \( f(x) \) for which \( g(f(x)) \) is not zero the elements in the domain of \( g(x) \) for which \( g(f(x)) \) is defined the elements in the domain of \( g(x) \) for which \( g(f(x)) \) is not zero
Álgebra United States Mar 21, 2025
For which pair of functions is \( (f \circ g)(x)=x \) ? \( f(x)=\frac{x^{2}}{x} \) and \( g(x)=\frac{1}{x} \) \( f(x)=\frac{x-2}{3} \) and \( g(x)=\frac{2}{x} \) \( f(x)=\frac{1}{2} x-2 \) and \( g(x)=\frac{1}{2} x+2 \)
Álgebra United States Mar 21, 2025
\[ f(x) = 3x + 4 \quad g(x) = 2x - 3 \] \[ \text{Encontre: } f \circ g(x) \]
Álgebra Angola Mar 21, 2025

Latest Algebra Questions

Which description best explains the domain of \( (g \circ f)(x) \) ? the elements in the domain of \( f(x) \) for which \( g(f(x)) \) is defined the elements in the domain of \( f(x) \) for which \( g(f(x)) \) is not zero the elements in the domain of \( g(x) \) for which \( g(f(x)) \) is defined the elements in the domain of \( g(x) \) for which \( g(f(x)) \) is not zero
Álgebra United States Mar 21, 2025
For which pair of functions is \( (f \circ g)(x)=x \) ? \[ f(x)=x^{2} \text { and } g(x)=\frac{1}{x} \] \( f(x)=\frac{2}{x} \) and \( g(x)=\frac{2}{x} \) \( f(x)=\frac{x-2}{3} \) and \( g(x)=2-3 x \) \( f(x)=\frac{1}{2} x-2 \) and \( g(x)=\frac{1}{2} x+2 \)
Álgebra United States Mar 21, 2025
For which pair of functions is \( (f \circ g)(x)=x \) ? \( f(x)=\frac{x^{2}}{x} \) and \( g(x)=\frac{1}{x} \) \( f(x)=\frac{x-2}{3} \) and \( g(x)=\frac{2}{x} \) \( f(x)=\frac{1}{2} x-2 \) and \( g(x)=\frac{1}{2} x+2 \)
Álgebra United States Mar 21, 2025
1-ALGEBRAIC EXPRESSIONS Gr10 In the following expressions, TWENTY possible solutions are provided for the instruction: Pactorise full or simpliti. FIVE of the fiventy solutions are algebraically correct and the rest are neolred. 4. Identify and CIRCLE OR CROSS the LETTER that represent solutions with correct (15) answers. 2 From the FIFTEEN incorrect solutions, identify and underline the mistakes made in EACH. 3. Correct the mistakes made in EACH incorrect solution.
Álgebra South Africa Mar 21, 2025
Guia \( 20 \quad 03 \quad 25 \) 1 factoriza 10s. Sig. trinomios Cuadros Perfectos 1) \( x^{2}-8 x+16 \) \[ x^{2}-8 x+16=(x-4)^{2} \] 2) \[ \begin{array}{l} x^{2}-20 x+100 \\ x^{2}-20 x+100=(x-10)^{2} \end{array} \] 3) \[ \begin{array}{l} x^{2}-24 x+144 \\ -2.12=-24 \\ x^{2}-24 x+144=(x-12)^{2} \end{array} \] 4) \( 25 x^{2}-18 x+81 \)
Álgebra Mexico Mar 21, 2025
Anterior Próximo
¡Prueba Premium ahora!
¡Prueba Premium y hazle a Thoth AI preguntas de matemáticas ilimitadas ahora!
Quizas mas tarde Hazte Premium
Estudiar puede ser una verdadera lucha
¿Por qué no estudiarlo en UpStudy?
Seleccione su plan a continuación
Prima

Puedes disfrutar

Empieza ahora
  • Explicaciones paso a paso
  • Tutores expertos en vivo 24/7
  • Número ilimitado de preguntas
  • Sin interrupciones
  • Acceso completo a Respuesta y Solución
  • Acceso completo al chat de PDF, al chat de UpStudy y al chat de navegación
Básico

Totalmente gratis pero limitado

  • Solución limitada
Bienvenido a ¡Estudia ahora!
Inicie sesión para continuar con el recorrido de Thoth AI Chat
Continuar con correo electrónico
O continuar con
Al hacer clic en "Iniciar sesión", acepta nuestros términos y condiciones. Términos de Uso & Política de privacidad